Apparatus and methods for image and signal processing

ABSTRACT

An apparatus and methods for efficiently processing signal and image data are described. The invention provides a representation of signal and image data that can be used as a figure of merit to compare and characterize different signal processing techniques. The representation can be used as an intermediate result that is may be subjected to further processing, and/or may be used as a control element for processing operations. As a provider of an intermediate result, the invention can be used as a step in processes for the transduction, storage, enhancement, refinement, feature extraction, compression, coding, transmission, or display of image, audio and other data. The invention improves manipulation of data from intrinsically unpredictable, or partially random sources. The result is a concise coding of the data in a form permitting robust and efficient data processing, a reduction in storage demands, and restoration of original data with minimal error and degradation. The invention provides a system of coding source data derived from the external environment, whether noise-free or contaminated by random components, and regardless of whether the data are represented in its natural state, such as photons, or have been pre-processed.

This invention was made with U.S. Government support under Grant No. EY03785, awarded by the National Institutes of Health (U.S.P.H.S.). TheU.S. Government may have certain rights to this invention.

RELATED APPLICATION

This application claims priority from applicants' co-pending U.S.provisional application entitled "Methods and Devices for SignalProcessing with Attribution, Phase Estimation, Adaptation, andQuantization Capabilities", bearing provisional application number60/054,399, filed Jul. 31, 1997, and incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention is generally related to the field of analog anddigital signal processing, and more particularly, to apparatus andmethods for the efficient representation and processing of signal orimage data.

2. Description of the Prior Art

FIG. 1 is a block diagram of a typical prior art signal processingsystem 100. As shown in the figure, such systems typically include aninput stage 102, a processing stage 104, an output stage 106, and datastorage element(s) 108.

Input stage 102 may include elements such as sensors, transducers,receivers, or means of reading data from a storage element. The inputstage provides data which are informative of man-made and/or naturallyoccurring phenomena. The informative component of the data may be maskedor contaminated by the presence of an unwanted signal, which is usuallycharacterized as noise. In some applications, an input element may beemployed to provide additional control of the input or processing stagesby a user, a feedback loop, or an external source.

The input data, in the form of a data stream, array, or packet, may bepresented to the processing stage directly or through an intermediatestorage element 108 in accordance with a predefined transfer protocol.Processing stage 104 may take the form of dedicated analog or digitaldevices, or programmable devices such as central processing units(CPUs), digital signal processors (DSPs), or field programmable gatearrays (FPGAs) to execute a desired set of data processing operations.Processing stage 104 may also include one or more CODECs(COder/DECcoders).

Output stage 106 produces a signal, display, or other response which iscapable of affecting a user or external apparatus. Typically, an outputdevice is employed to generate an indicator signal, a display, ahardcopy, a representation of processed data in storage, or to initiatetransmission of data to a remote site, for example. It may also beemployed to provide an intermediate signal for use in subsequentprocessing operations and/or as a control element in the control ofprocessing operations.

When employed, storage element 108 may be either permanent, such asphotographic film and read-only media, or volatile, such as dynamicrandom access memory (RAM). It is not uncommon for a single signalprocessing system to include several types of storage elements, with theelements having various relationships to the input, processing, andoutput stages. Examples of such storage elements include input buffers,output buffers, and processing caches.

The primary objective of signal or information processing system 100 isto process input data to produce an output which is meaningful for aspecific application. In order to accomplish this goal, a variety ofprocessing operations may be utilized, including noise reduction orcancellation, feature extraction, data categorization, event detection,editing, data selection, and data re-coding.

The design of a signal processing system is influenced by the intendeduse of the system and the expected characteristics of the source signalused as an input. In most cases, the performance efficiency required,which is affected by the available storage capacity and computationalcomplexity of a particular application, will also be a significantdesign factor.

In some cases, the characteristics of the source signal can adverselyimpact the goal of efficient data processing. Except for applications inwhich the input data are naturally or deliberately constrained to havenarrowly definable characteristics (such as a limited set of symbolvalues or a narrow bandwidth), intrinsic variability of the source datacan be an obstacle to processing the data in a reliable and efficientmanner without introducing errors arising from ad hoc engineeringassumptions. In this regard, it is noted that many data sources whichproduce poorly constrained data are of importance to people, such assound and visual images.

Conventional image processing methods suffer from a number ofinefficiencies which are manifested in the form of slow datacommunication speeds, large storage requirements, and disturbingperceptual artifacts. These can be serious problems because of thevariety of ways it is desired to use and manipulate image data, andbecause of the innate sensitivity people have for visual information.

Specifically, an "optimal" image or signal processing system would becharacterized by, among other things, swift, efficient, reliable, androbust methods for performing a desired set of processing operations.Such operations include the transduction, storage, transmission,display, compression, editing, encryption, enhancement, sorting,categorization, feature detection and recognition, and aesthetictransformation of data, and integration of such processed data withother information sources. Equally important, in the case of an imageprocessing system, the outputs should be capable of interacting withhuman vision as naturally as possible by avoiding the introduction ofperceptual distractions and distortion.

That a signal processing method should be robust means that its speed,efficiency, and quality (for example), should not depend strongly on thespecifics of any particular characteristics of the input data, i.e., itshould perform "optimally," or near that level, for any plausible input.

This is an important aspect because a common inadequacy suffered bysignal processing methods is their failure to be robust. JPEG-typemethods in imaging, for example, perform better for "photographic"images having gentle gradations in color and luminance than for graphicimages and others having sharp discontinuities. On the other hand, imagecompression methods such as those embodied in the GIF format performbest when an image has few of the complexities found in photographicimages. Similar examples may be cited with regard to processingoperations performed on audio and other classes of input data.

In part, conventional image processing methods lack robustness becausethere are an infinite number of possible images. Adding to this is thecomplication that in most situations, it is impossible to knowbeforehand exactly what features and complexities an image will possess.Thus, to describe an image entirely, one approach is to determine theluminance and color of every point in the image. However, the volume ofinformation needed to accomplish this task can exceed several megabytesfor a digital image of moderate size, making it burdensome to store,process, and transmit such information. Even then, the digitalrepresentation is an inexact record of the original image owing to thelimitations inherent in constructing binary value based representationsof continuous analog signals.

Information is lost in any discrete representation of continuous-valueddata because discrete sampling over any finite duration or area cannotcapture all of the variations in the source data. Similarly, informationis lost in any quantization process when the full range of values in thesource data cannot be represented by a set of discrete values.

In addition to difficulties imposed by the nature or implementation of aprocessing operation, other problems must be addressed whencontaminating noise sources mask or distort the component of an inputthat is assumed to represent a signal of interest. However, it is rarelyappreciated that there are other forms of randomness andunpredictability which cannot be defined legitimately as noise but whichare nonetheless the source of problems with regard to the quality androbustness of signal processing methods. These forms of unpredictabilitymay be considered in terms of intrinsic randomness and ensemblevariability. Intrinsic randomness refers to randomness that isinseparable from the medium or source of data. The quantal randomness ofphoton capture is an example of intrinsic randomness.

Ensemble variability refers to any unpredictability in a class of dataor information sources. Data representative of visual information has avery large degree of ensemble variability because visual information ispractically unconstrained. Visual data may represent any temporalseries, spatial pattern, or spatio-temporal sequence that can be formedby light. There is no way to define visual information more precisely.Data representative of audio information is another class of data havinga large ensemble variability. Music, speech, animal calls, wind rustlingthrough the leaves, and other sounds share no inherent characteristicsother than being representative of pressure waves. The fact that peoplecan only hear certain sounds and are more sensitive to certainfrequencies than to others is a characteristic of human audio processingrather than the nature of sound. Examples of similarly variable classesof data and information sources can be found throughout nature and forman-made phenomena.

The unpredictability resulting from noise, intrinsic randomness, andensemble variability, individually and in combinations, makes itdifficult and usually impossible to extract the informative or signalcomponent from input data. Any attempt to do so requires that a signaland noise model be implicitly or explicitly defined. However, no signaland noise model can be employed which is able to assign with absoluteconfidence a component of input data to the category of informativesignal as opposed to uninformative noise when there is any possibilitythat the noise, intrinsic randomness, or ensemble variability sharecharacteristics.

A signal and noise model is implicitly or explicitly built into a signalprocessing operation in order to limit the variability in its output andto make the processing operation tractable. Signal processors generallyimpose some form of constraint or structure on the manner in which thedata is represented or interpreted. As a result, such methods introducesystematic errors which can impact the quality of the output, theconfidence with which the output may be regarded, and the type ofsubsequent processing tasks that can reliably be performed on the data.

An often unstated but significant assumption employed in signalprocessing methods is that source data can be represented orapproximated by a combination of symbols or functions. In doing so, suchprocessing methods essentially impose criteria by which values andcorrelations in an input are defined or judged to be significant. Asignal processing method must embody some concept of what is to beregarded as signal. However, the implicit or explicit presumption that acertain set of values or certain kinds of correlation can be use toprovide a complete definition of a signal is often unfounded and leadsto processing errors and inefficiencies. By defining a signal in termsof a set of values or correlations, a processing method is effectivelyassigning all other values and correlations to the category of noise.Such an approach is valid only when it is known that: 1) the informationsource that the input data represents takes on only a certain set ofvalues or correlations; and 2) noise or randomness in the input datanever cause the input to take on those values or correlations by chance.Conditions of this sort are rare at best and arguably never occur inreal life. These conditions are certainly not true for visual, audio, orother information sources which have an unconstrained ensemblevariability. For such classes of data, a finite set of values orcorrelations is insufficient to completely cover the range ofvariability that exists. As a result, some values or correlations whichare representative of an information source will be inevitable anderroneously assigned to the category of noise. It should be noted thatthe inventive method herein does not presume such a set of specificvalues or correlations.

To further illustrate some of the limitation of signal and noise modelsin general, we discuss in this section several processing techniqueswhich are found in the field of image processing. Among conventionalimage and signal processing techniques are histogram methods, predictivecoding methods, error coding methods, and methods which represent datain terms of a set of basis functions such as JPEG, MPEG, andwavelet-based techniques.

Histogram methods are based on categorizing the luminance and colorvalues in an image, and include the concept of palettes. A histogram isrelated to a probability density function which describes how frequentlyparticular values fall within specified range limits. Histogram methodsare used to quantize source data in order to reduce the number ofalternative values needed to provide a representation of the data. Inone form or another, a histogram method has been applied to everydigital image that has been derived from continuous-valued source data.Histogram methods are also used for aesthetic effect in applicationssuch as histogram equalization, color re-mapping, and thresholding.

However, a disadvantage of histogram techniques is that the processingscheme used to implement such methods must determine which ranges ofvalue and color are more important or beneficial than others. Thisconflicts with the fact that the distribution of values in an imagevaries dramatically from one image to the next. Similarly, the numberand location of peaks and valleys in a histogram varies significantlybetween images. As a result, histogram methods are computationallycomplicated and produce results of varying degrees of quality fordifferent kinds of images. They also tend to produce an output havingnoticeable pixelation and unnatural color structure.

Predictive coding methods attempt to compensate for some of thelimitations of histogram methods by considering the relationship betweenthe image values at multiple image points in addition to the overalldistribution of values. Predictive coding techniques are suited to datahaving naturally limited variability, such as bi-tonal images. Suchmethods are an important part of the JBIG and Group 3/4 standards usedfor facsimile communications. However, for more complicated image datasuch as multi-level grayscale and full color images, predictive codingmethods have not been as effective.

Predictive coding techniques are based on the hypothesis that there arecorrelations in image data which can be used to predict the value of animage at a particular point based on the values at other points in theimage. Such methods may be used to cancel noise by ignoring variationsin an image that deviate too significantly from a predicted value. Suchmethods may also be used in image compression schemes by coding an imagepoint only when it deviates significantly from the value predicted.

However, one of the problems encountered in predictive coding is thedifficulty in deciding that a particular deviation in an image is animportant piece of information rather than noise. Another source ofdifficulty is that correlations in an image differ from place to placeas well as between images. At present, no conventional predictive codingmethod has employed a sufficiently robust algorithm to minimizeprocessing errors over a realistic range of images. As a result,conventional predictive coding methods tend to homogenize variationsbetween images.

Error coding methods extend predictive methods by coding the errorbetween a predicted value and the actual value. Conventional errorcoding methods tend to produce a representation of the input data inwhich small values near zero are more common than larger values.However, such methods typically do not reduce the total dynamic rangefrom that of the input data and may even increase the range. As aresult, error coding methods are susceptible to noise and quantizationerrors, particularly when attempting to reconstruct the original sourcedata from the error-coded representation. In addition, since errorcoding is an extension of predictive coding, these two classes ofmethods share many of the same problems and disadvantages.

Representation of data using a set of basis functions is well known,with Fourier techniques being perhaps the most familiar. Other transformmethods include the fast Fourier transform (FFT), the discrete cosinetransform (DCT), and a variety of wavelet transforms. Therationalization for such transform methods is that the basis functionscan be encoded by coefficient values and that certain coefficients maybe treated as more significant than others based on the informationcontent of the original source data. In doing so, they effectivelyregard certain coefficient values and correlations of the sort mimickedby the basis functions as more important than any other values orcorrelations. In essence, transform methods are a means of categorizingthe correlations in an image. The limitations of such methods are aresult of the unpredictability of the correlations. The variations inluminance and color that characterize an image are often localized andchange across the face of the image. As a result, FFT and DCT basedmethods, such as JPEG, often first segment an image into a number ofblocks so that the analysis of correlations can be restricted to a smallarea of the image. A consequence of this approach is that bothersomediscontinuities can occur at the edges of the blocks.

Wavelet-based methods avoid this "blocking effect" somewhat by usingbasis functions that are more localized than sine and cosine functions.However, a problem with wavelet-based methods is that they assume that aparticular function is appropriate for an image and that the entireimage may be described by the superposition of scaled versions of thatfunction centered at different places within the image. Given thecomplexity of image data, such a presumption is often unjustified.Consequently, wavelet based methods tend to produce textural blurringand noticeable changes in processing and coding quality within andbetween images.

To address some of the problems arising from the complexity of images asan information source, a number of attempts have been made toincorporate models of human perception into data processing methods.These are based on the belief that by using human visual capabilities asa guide, many of the errors and distortions introduced during processingcan be rendered inconsequential. In essence, use of human perceptualmodels provides a basis for deciding that some visual information ismore important than other information. For example, television andseveral computer image formats explicitly treat luminance information asmore important than color information and preferentially devote codingand processing resources to grayscale data. While this approach showspromise, there is no sufficiently accurate model of human perceptioncurrently available to assist in processing image data. As a result,attempts to design processes incorporating such models have resulted inimages that are noticeably imperfect.

What is desired and needed are apparatus and methods for the processingof general signal and image data which are more efficient thanconventional approaches. In particular, signal and image processingapparatus and methods are desired which are less computationally complexand have reduced data storage requirements compared to conventionalmethods. Apparatus and methods for reconstructing signals and imagesfrom processed data without the degradation of signal or image qualityfound in conventional approaches are also desired.

The present invention provides such apparatus and methods.

SUMMARY OF THE INVENTION

The present invention is directed to apparatus and methods forefficiently processing signal and image data. The inventive methodprovides a representation of signal and image data which can be used asan end product or as an intermediate result which is subjected tofurther processing. As an end product, the data representation providesa figure of merit that can be used to compare and characterize differentsignal processing techniques, or as a control element for causingadaptation of a processing operation. As a provider of an intermediateresult, the method can be used as a step in processes for thetransduction, storage, enhancement, refinement, feature extraction,compression, coding, transmission, or display of image data. In thiscontext, the inventive method significantly reduces the computationaland data storage requirements of conventional signal processing methods.The invention provides improved methods of manipulating data fromintrinsically unpredictable, or partially random sources to produce aconcise coding of the data in a form that allows for more robust andefficient subsequent processing methods than is currently possible, areduction in storage demands, and restoration of the original data withminimal error and degradation. The invention provides a system of codingsource data derived from the external environment, whether noise-free orcontaminated by random components, and regardless of whether the data isrepresented in its natural state, such as photons, or has beenpre-processed.

Other features and advantages of the invention will appear from thefollowing description in which the preferred embodiments have been setforth in detail, in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a typical prior art signal processingsystem.

FIG. 2 is a block diagram showing the processing of a set of input datainto an output according to the method of the present invention, withthe processing operation(s) represented as a two-stage operation.

FIG. 3 is a block diagram showing the relationships between the inputdata set, processing function, uncertainty operator, uncertainty signal,and the signal estimate, in accordance with the present invention.

FIG. 4 is a block diagram showing a signal estimate operated on by anuncertainty task or bias to generate the uncertainty signal, subjectedto further processing steps, and then operated on by the inverse of thetask to obtain a new estimate of the signal.

FIG. 5 is a block diagram illustrating how the present invention may beused to generate a figure of merit for purposes of monitoring a signalprocessing operation.

FIG. 6 is a block diagram illustrating how the inventive uncertaintysignal may be used as an intermediate form of processed data to replacea signal representation prior to application of additional processingoperations.

FIG. 7 is a block diagram illustrating how the inventive uncertaintysignal may be used to control the operation of processes and/orprocessing tasks.

FIG. 8 is a block diagram illustrating a second manner in which theinventive uncertainty signal may be used to control the operation ofprocesses and/or processing tasks.

FIG. 9 is a block diagram illustrating how the inventive signalprocessing methods may be used to perform data emphasis and de-emphasis.

FIG. 10 is a block diagram illustrating the use of the inventive signalprocessing methods for constructing an uncertainty process from apre-existing or hypothetical signal or data processing operation.

FIGS. 11a and 11b are flow charts showing primary signal processingsteps implemented to determine the uncertainty filter and uncertaintytask from an I/O analysis of a processing scheme according to the methodof the present invention.

FIG. 12 is a block diagram illustrating methods of implementing theattribution process, uncertainty process, uncertainty task, and relevantinverse processes in accordance with the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides a signal processing method and apparatusimplementing such method, which method and apparatus are advantageouslyapplicable to any type of data represented in a format suitable forapplication of the disclosed processing operations. Without limitation,the data can include both digital data and analog data, and datarepresentative of visual images, audio signals, and radar signals. Whileportions of the following description refer to or may be directed tovisual image data processing, it is important to appreciate that thepresent invention is not limited to use with such data, and thatreference to such data is for purposes of example only. Similarly, it isnoted that the mathematical description of the inventive method uses theform of a generalized frequency notation. The generalized frequency maybe read as a temporal, spatial, or spatio-temporal frequency, and isutilized because the fundamental processing methodology does not differfor time, space, or space-time. Temporal processing simply requires thatthe conditions of causality be satisfied. The use of frequency domainnotation should not be taken to mean that data need conversion into thefrequency domain for processing; rather, the frequency domain termsshould be thought of symbolically. It is often preferable to process thedata as it arrives in time and space using circuits, for example, of thetype described in the copending provisional application. This is one ofthe advantages of the inventive method, which performs what might beburdensome computations in other processing methods simply and quicklyby using such circuits.

Applicants have come to recognize that the commonly made assumption inthe prior art that some kinds of information or correlations are moreimportant than others is the source of many of the problems which arisein the processing of complicated data sources. This assumption ismanifested both in the choice of which signal processing method(s) toapply to the data and is also the basis for the operations performed bymost conventional signal processing schemes. For example, histogrammethods essentially categorize value ranges in terms of visualimportance for specific images. In one way or another, predictivecoding, error coding, and basis function methods implicitly orexplicitly assume that certain kinds of variations in image data aremore significant than others. Such methods are based on ad hocengineering assumptions even if in some cases they are partly supportedby a theoretical or empirical model such as a model of human perception.As a result, such methods are a source of procedural bias in the dataprocessing because they introduce systematic errors that arise from theprocessing method, rather than being a result of the inherentcharacteristics of the source data.

The introduction of such systematic errors may be thought of as theintroduction of systematic misinformation. Thus, most conventionalprocessing methods impose specific constraints on the data that resultin inefficient and sometimes erroneous interpretations and manipulationsof the data.

That some conventional processing methods possess inadequacies orinefficiencies does not mean they are without merit for particularapplications. However, the presumptions implicit in such methodsrestrict their versatility and also limit the processing operationswhich can be performed on the data while maintaining a desired degree ofconfidence in the result. For example, it is probably inappropriate toperform a fingerprint recognition operation on a blocky JPEG image aserrors introduced by the DCT quantization result in reduced efficiencyand can lead to misidentification. Similar arguments can be made aboutother methods that presume that some information is more important thanother information, or that certain characteristics of a set of datadetermine whether it should be assigned to signal or to noise. Once sucha method is applied to source data, the range of valid operations thatcan be subsequently performed becomes limited.

One advantage of the inventive method is that discrete sampling methodscan be employed in such a manner as to minimize information loss.Moreover, the inventive method provides ways in which continuous-valuedrepresentations of source data can be generated from a discreterepresentation.

A significant feature of the inventive method is that it creates, frominput data and an implicit or explicit signal and noise model, a metricof confidence that has characteristics superior to those of aconventional representation of a signal: it may be used in place of asignal representation in many signal processing operations; it may beused to control the quality and efficiency of processing operations; andit may be used to characterize existing or hypothetical processingoperations. Consequently, the inventive method can be used to controland quantify the errors that may be introduced by the imposition of asignal and noise model.

The signal and noise concept is so ingrained that it is unquestioned andits limits unexplored. In arriving at the present invention, applicantshave had to reconsider the signal and noise concept, which they havecome to realize is not incorrect but rather incomplete. The assignmentof aspect of input data to either signal or noise is generally attendedby uncertainty as to the confidence that should be placed on such anassignment. The inventors have realized that such uncertainty can berepresented in a manner that stands apart from the representation of asignal and a representation of a noise. That uncertainty signalrepresents the power in the input that cannot be attributed to eithersignal or noise alone; i.e., it serves as a metric of confidence.Applicants have also come to realize that the uncertainty signalrepresents the information source that gave rise to the input data in acompact manner that may be used both in place of a representation of asignal and as a control signal for controlling information processingoperations.

In considering the shortcomings of conventional signal and informationprocessing methods (such as those described above), applicants realizedthat a reliable and efficient signal processing method should havecertain characteristics. These include, but are not limited to:

(1) the method should embody a minimum of ad hoc assumptions and sourcesof procedurally introduced bias to minimize systematic errors andmaximize versatility;

(2) the method should be computationally simple and efficient;

(3) the method should be reliable and robustly applicable to complexdata sources;

(4) the method should provide a means of minimizing noise and randomnessin source data without requiring detailed knowledge of which datacomponents are informative and which are contamination;

(5) the method should introduce a minimum amount of distortion;

(6) the method should allow for input data to be quantized and sortedwith minimal signal deterioration;

(7) the method should allow for a high degree of data compression;

(8) the method should allow processed data to be efficiently transmittedto remote locations;

(9) the method should be able to adapt to changes in the source data toreduce processing errors and inefficiencies; and

(10) the method should be able to be implemented using either analogand/or digital techniques as is appropriate for a particularapplication.

In considering these requirements, applicants questioned the traditionalconcept that some information can be classified as more important thanother information. With regards to image processing, applicantsreconsidered the assumption that the luminance and color values in animage should be considered the raw information. Applicants realized thatluminance and color do not provide the most efficient, robust andreliable information about an image which can be processed to extractdesired information about the data. This realization and its extensionto other types of information sources and data types has resulted in anumber of concepts that help form the basis of the present invention.

The Ambiguous Component of the Input Data

Every signal and information processing method strives to produce someresult from a set of inputs. The input may be, and commonly is,described as having two components: a signal component that contains theinformation or message, and a noise component that reflects distortionsof the signal component and contamination in the form of randomvariations (random noise) and crosstalk, for example. The presentinvention recognizes that the initial step of defining an input ashaving a signal component and a noise component has vast implicationsbecause it imposes a particular model on the data. It essentiallyrequires that all of the data be categorized as either signal or noise,with the associated ramifications regarding presumed magnitude and phaserelationship(s) between a set of signal data and other signal data,signal data and noise data, and a set of noise data and other noisedata. The bias of the model choice introduces limitations on theprecision with which the data can be processed while maintaining a"bright line" which differentiates signal from noise.

Given a signal and noise model, the input may be written as X(v), thesignal as S(v), and the noise as N(v), where the capital lettersdesignate a frequency domain representation (e.g., the Fouriertransform) and the parameter v represents a generalized frequency(typically a temporal, spatial, or spatio-temporal frequency). In such asituation, the input, as the sum of signal and noise, may be written as:

    X(v)=S(v)+N(v).

In using such a data model, the input data X(v) is known, and a modelfor the noise contribution N(v) is assumed. Based on these terms, arepresentation of the signal S(v) is determined.

However, despite the wide-spread convention of representing data interms of signal and noise components, applicants realized that there isa more efficient and versatile way of processing input data,particularly data arising from complicated sources. One motivation forthe present invention is that the assumption that input data can bedecomposed into signal and noise components is incongruous with thereality of complicated information sources, as one can rarely, if ever,precisely define the signal components of a data set from a prioriknowledge. Attempts to impose a definition of signal in a particularprocessing scheme implicitly defines the noise, introduces systematicerror, and restricts the type of processing operations which canreliably be performed on the data. For example, the conventional imageprocessing methods described above presume that some aspect(s) orcharacteristics of the input data are more significant than others,e.g., value ranges or types of correlations. These methods inherentlydefine the signal component and thus can result in the kind ofprocessing limitations described.

Thus, the present invention realizes that the "decision," implicitly orexplicitly, as to what is signal and what is noise has introducedinefficiencies into conventional signal processing schemes and renderedthem sub-optimal. Instead, what is desired is a method of"interpretation" which does not introduce these disadvantages. Thus,using this approach, the present invention provides apparatus andmethods of representing input data from complex sources in terms ofmeasures of ambiguity and uncertainty, instead of in terms of signal andnoise. These methods, and this kind of data representation, have severaladvantages over the traditional signal and noise approach.

The concept of interpretation is in some ways similar to that ofestimation. Estimation theory is a starting point, but this should notbe construed as a limitation on the scope of the present invention. Forexample, the explicit use of noise terms in the following development isincluded for generality and should not be taken to mean that the presentinvention is limited to noisy data sources. In the classic signalestimation problem, the goal is to produce the best possible estimate ofa signal component from an input. Representing the estimated signal asS'(v), the operation may be represented generically as: X(v)-S'(v).Producing an estimate of the signal also produces an estimate of thenoise component:

    N'(v)=X(v)-S'(v).

However, just what processing operation should be performed to producethe estimate depends on how one defines "best" and what constraints areimposed on the characteristics of signal and noise.

The problem is that when there is a possibility that signal and noisecomponents of the input data could be confused, or when a preciselyaccurate definition of the signal or noise is not possible (as is thecase for many complex information sources, such as visual images), thereis a possibility that the estimation process will misinterpret or ignoresome portion of the informative content of the input data. This meansthat there will be some ambiguity.

This possible "misinterpretation" arises because some correlations inthe input data could be attributed either to the signal or to the noisecomponent, instead of confidently assigned to one or the other. Indeed,any signal estimation process, linear or not, may be described as one inwhich correlations in the input are weighted according to how likely itis that those correlations are informative of the message rather than ofcontamination, given some prior expectations or definitions concerningthe signal and noise and some weighting criterion.

This potential ambiguity may be understood in terms of correlationsbetween the supposed signal and noise components. Correlations are oftendiscussed in terms of amplitude and phase correlations. The input datamay be written as:

    X(v)=|X(v)|exp(iθ.sub.X (v)),

where |X(v)| is the amplitude spectrum and θ_(X) (v) is the phasespectrum of that data. Similarly:

    S'(v)=|S'(v)|exp(iθ.sub.S' (v)) and N'(v)=|N'(v)|exp(iθ.sub.N' (v)).

Note that this formulation does not presume a linear relationshipbetween the input data and the estimates, and is a valid mathematicalstatement independent of the processing method. Using the abovenotation, the power associated with correlations in the input data maybe represented by:

    |X(v)|.sup.2 =|S'(v)|.sup.2 +2|S'(v)∥N'(v)|cos(θ.sub.S' (v)-θ.sub.N' (v))+|N'(v)+|N'(v)|.sup.2.

The squared amplitude spectra may be read as power spectra. The equationillustrates that the input power may be represented as the sum of thepower in the estimated signal plus the power in the estimated noise,plus a cross term (the middle term) which represents the remainingpower. This remaining power is the power in the input that cannot beaccounted for by the estimated signal and noise viewed independently ofeach other. In one sense it represents the power that cannot beattributed to either the signal alone or the noise alone with sufficientconfidence, based on the signal and noise model adopted. It is theambiguous power due to correlations between the signal and noiseestimates, and is thus a measure of the limitations or imprecision ofthe model used to assign the input data to either signal or noise.

As used herein, the aspects of the input data that cannot be ascribedwith sufficient confidence to signal alone or to noise alone is termedthe "ambiguous" component of the input. Note that, in the conventionalsignal and noise paradigm, the ambiguous component is not a separateentity, i.e., the input data is fully described by the signal and noiseestimates, X(v)=S'(v)+N'(v). The ambiguous component preferentiallyrepresents the correlations in the input that are least predictable. Theambiguous component has largely been ignored in conventional signal andinformation processing because it is believed to represent the aspectsof source data that are too uncertain to be a reliable source ofinformation. Based on a recognition of the significance of the ambiguouscomponent of input data, the present invention recognizes thatrepresenting or extracting this component by performing an operation onthe input data, many of the problems associated with other signalprocessing methods could be avoided and/or controlled.

Thus, the present invention recognizes that application of a signal andnoise model to the processing of input data introduces a source of errorin the processing because it requires that each piece of data beassigned to either signal or noise. However, there is some input datapower that is not assigned to either signal or noise, i.e., theambiguous component. In conventional processing schemes, this input datapower is ignored, with the result that some information contained in theinput data is lost. However, the present invention provides a method forextracting this previously lost information and utilizing it to improveprocessing of the data.

In determining an operation to perform on input data to extract theambiguous component, applicants were guided by the previously identifiedcriteria for reliable and efficient processing. By implementing a methodbased on a minimum number of assumptions and which minimizes datadistortions, the present invention can satisfy many, if not all, of thecriteria. Further, the present invention recognizes that imposing aminimum of assumptions as to the nature of the data has a direct bearingon how distortions could be minimized.

Estimation of the Ambiguous Component

To introduce the minimum number of assumptions regarding the form ornature of the input data, it is instructive to return to the idea thatsome correlations in input data may be more important than others andthat one can rely on such a characterization before the data isprocessed. In image data, for example, query whether the edges should betreated as more significant than smooth gradations. It is arguable thatedges are perceptually a more significant feature, however, toincorporate the concept of an edge in a processing method, it isnecessary to define the characteristics of an edge. This is a moredifficult task than might be suspected. What most would agree to be anedge in an image is typically a gradation of intensity or color over anarrow region rather than an abrupt transition. It is, of course,possible to define an edge as a feature that changes by a certain amountwithin a certain area, but this ignores the fact that the gradationcould take the form of a step or a ramp or other transition function. Inaddition, one must also be aware that an edge is not always the mostperceptually significant feature. For example, whereas an edge might beimportant in an image of buildings, it may not be in an image of alandscape at sunset. In order to assure optimal processing versatilityit is desirable to adopt a measure of importance that is valid not onlywithin an image but also between images of different kinds.

Transitions and variations in source data are partly definable by phasecorrelations. Phase is not an absolute metric because it refers to therelationship between different parts of the data. In images, forexample, phase information indicates how certain features or transitionsare located with respect to others. Thus, to define a set ofcorrelations as more important than others would require a referencepoint; e.g., where a camera was pointed or the time when data wereacquired. However, for complicated data sources, there is no way todefine reference points so that the input data are likely to haveparticular phase characteristics, particularly if the input data containrandom disturbances. Multiple exposures of a piece of film, for example,will tend to produce a gray blur because there is no likelihood thatcertain image features will line up in a particular way with respect tothe camera.

Thus, to minimize the number of assumptions and maximize versatility,the present invention recognizes the desirability of processingdifferent kinds of phase correlations in a similar manner. An advantageof this approach is that specialized processing operations which embodyassumptions about the importance of different kinds of phasecorrelations can be performed subsequently without constraining thetypes of other possible processing operations.

Processing methods that have a minimal impact on the phasecharacteristics of a set of input data are linear. The only phasedistortions necessarily introduced by such methods are those that arisefrom the fact that processing can only be performed on data that hasalready been acquired. Linear processes that introduce the minimalamount of phase distortion allowed by the principle of causality aretermed "minimum-phase processes". Further information regardingminimum-phase processes may be found in the reference Kuo, F. F. (1966)Network Analysis and Synthesis. 2nd. Ed. Wiley & Sons: New York.

In the purely spatial case, as for still images, where time is not afactor, the inventive processing method will introduce zero phasedistortion. In temporal and spatio-temporal cases in which an output isdesired in real time, the inventive processing method will meet at leastthe criteria for minimum-phase processes as the characteristics of suchprocesses are understood by those of skill in the art of signalprocessing. In cases in which data are stored before processing, a delayequivalent to a phase distortion is introduced and the phasecharacteristics of the inventive method need not be constrained. Notethat the technique of Wiener-Hopf spectrum factorization may also beused to define the phase characteristics of the inventive method tosatisfy the causality constraint. Further details regarding Weiner-Hopfspectrum factorization may be found in the reference Pierre, D. A.(1986) Optimization Theory with Applications. Dover: New York.

Note that the conclusion that a desirable processing operation should belinear is independent of whether it is desired to estimate the signaland noise components from input data or represent the ambiguity. Thus,in the linear signal estimation problem, the estimated signal may bewritten as:

    S'(v)=W(v)X(v),

where W(v) represents a processing operation having an amplitudespectrum |W(v)| and a phase spectrum θ_(W) (v). Similarly,

    |S'(v)=|W(v)|X(v)|| and θ.sub.S' (v)=θ.sub.W (v)+θ.sub.X (v),

Recognizing that any phase distortion introduced in-processingintrinsically has nothing to do with the signal processing problem, onecan imagine a non-causal, zero-phase operation, X'(v), which wouldproduce the result:

    S'(v)=|W(v)|X'(v),

where X'(v)=X(v)exp(iθ_(W) (v)).

Consequently, the effective noise estimate would be:

    N'(v)=(1-|W(v)|)X'(v)).

The magnitude of the ambiguous power component may therefore be writtenas:

    2|S'(v)∥N'(v)|=2|W(v)|(1-.vertline.W(v)|)|X'(v)|.sup.2,

Note that |X'(v)|² =|X(v)|².

Despite the fact that in some sense any processing operation may beconsidered a signal estimation process, it is more common to think of aprocessing operation as something that performs a task on a signal orsignal estimate that is produced by a sub- or pre-processor. Thedistinction between the notions of signal estimation and task arisesfrom the conventional view of signal and noise.

FIG. 2 is a block diagram showing the processing of a set of input data,X, into an output, Y according to the present invention, with theprocessing operation(s) represented as a two-stage operation, in thiscase a combination of a signal estimation operation and a processingtask. FIG. 2a shows the input X(v) being processed by a set ofprocessing operations represented by box 200 to produce an output, Y(v).As shown in FIG. 2b, the processing operations of box 200 may berepresented as a combination of a signal estimation process W (box 202),which operates on X(v)=S(v)+N(v) to produce a signal estimate, S'(v),followed by a processing task, G_(f), which operates on the signalestimate to produce the output Y(v)=G_(f) S'.

The estimation stage (box 202) may be characterized as a universalpre-processor. For example, data which are input to an array of separateprocessors performing signal processing operations may be represented interms of a single, shared signal estimation process and an array ofprocessing tasks subserving the various operations.

Note that in many cases, what might be regarded as input data may alsobe regarded as a signal estimate in the sense of a signal andinformation processing operation. For example, a digital representationof a photograph might be considered an estimate of the actual luminanceand spectral components of the real world. It is not intended to limitthe scope of the invention to cases in which input data may beconsidered noisy in the conventional sense. The term signal estimaterefers to any data which may be regarded to be representative of aninformative source.

One goal of the present invention is to produce a representation of theambiguous component of the input data in a manner that is robust in thesense of being applicable to any possible input. By inspecting thepreceding equation for the power of the ambiguous component, andrecognizing that |X'(v)|² =|X(v)|², the present inventors recognizedthat the linear operation:

    D'(v)=U(v)X(v),

where U(v) is a zero-phase or minimum-phase process having an amplitudespectrum given by: ##EQU1## would accomplish that goal. The variableD'(v) denotes a result obtained from the input data that indicates theambiguity in the input data given the implicit signal and noise modelembodied in the processing operation W(v). As used herein, U(v) andD'(v) are termed the uncertainty process and signal, respectively. Notethat W(v) satisfies the relation |W(v)|<1. If required, processingfunction W(v) should be scaled or normalized to satisfy thisrelationship. Note that the power of the uncertainty signal is one-halfthe ambiguous component of power. The factor of one-half was chosen soone could imagine that the ambiguous component of power is split evenlybetween uncertainty associated with a signal estimate and uncertaintyassociated with a noise estimate. Note that this choice of scalingshould not be taken so as to limit the inventive method. The uncertaintyprocess is constrained only by its frequency dependence.

As for any processing operation, the uncertainty process may berepresented as a combination of a signal estimation stage and aprocessing task as noted. Thus U(v) may be represented as:

    U(v)=W(v) G.sub.U (v)

so that:

    D'(v)=U(v)X(v)=G.sub.U (v)W(v)X(v)=G.sub.U (v) S'(v)

As used herein, G_(u) is termed the uncertainty task, and the process itrepresents has an amplitude spectrum characterized by: ##EQU2## Tomaximize versatility of the uncertainty task, it may have zero- orminimum-phase characteristics, although other phase characteristics maybe appropriate as noted.

The uncertainty signal, D'(v), provides a concise indication of thequality or reliability associated with an implicit or explicitimposition of a signal and noise model by a processing operation. Itspoint-to-point value provides an estimate of the probable errorassociated with the signal and noise measures. Its root-mean-squarevalue (or any equivalent), judged against that of the input data,provides a measure of overall reliability of the estimation process. Theuncertainty signal is a stand-alone linear transformation of the inputdata. It may be produced from the representation of a signal or noise,but can also be produced directly from input data without having toproduce signal and noise estimates. In many cases of interest, includingthe visual case, the signal and noise estimates, if desired, may beproduced via a linear transformation of the uncertainty signal insteadof the original input data. Thus, the uncertainty signal can be used asa substitute for the signal estimate as the primary representation ofthe input data. An advantage to this representation is that the inputdata will tend to be represented with less power and a narrower dynamicrange. This aspect of the uncertainty signal is advantageous for datacompression applications.

FIG. 3 is a block diagram showing the relationships between the inputdata set X, the processing function W, the uncertainty operator U,uncertainty signal D', and the signal estimate S', in accordance withthe present invention. As shown in FIG. 3a, the input signal X(v) isoperated on by the processing function W(v) to form the signal estimateS'(v). The input data may also be operated on by the uncertaintyoperator U(v) to produce the uncertainty signal D'(v), as shown in FIG.3b. This process may also be represented as a combination of theprocessing function W(v) and an uncertainty processing task, G_(u), asshown in FIG. 3c. This two-stage approach has the advantage that boththe signal estimate and uncertainty signal are made available forsubsequent processing operations. Note that the uncertainty signal maybe obtained by operating on the output of an estimation process, or on arepresentation of a signal.

The uncertainty signal may also be used as an indicator of the qualityof a processing operation, although the uncertainty process and theuncertainty signal are even more versatile. The uncertainty processtends to preferentially report those aspects of an input which are mostunique and unexpectable; i.e., in terms of what is least predictable andmost uncertain with regard to an implicit or explicit signal and noisemodel. The uncertainty signal tends to have a more compact andpredictable dynamic range than typical signal data, and contains thesame information content as a signal estimate. It provides a measure ofthe root-mean-square error that can be expected in an estimation processor signal representation. It also provides a characterization of thephase properties of input data and/or a signal estimate without the needfor additional processing.

Typically, in designing a signal processing method, tasks such asfeature emphasis or de-emphasis, compression, process monitoring,feature detection or extraction, phase extraction, dynamic rangeoptimization, transmission and reception, and a variety of controlprocesses are treated as separate processes, with each performingspecific and unique operations on input data. However, thecharacteristics of the uncertainty signal demonstrate how the inventiveuncertainty process acts to unify and simplify such processing tasks.

Because it contains the same informative value as a signalrepresentation, many processing operations that might have beenperformed on a signal representation may instead be performed on theuncertainty representation with either zero or minimal loss ofinformative value. Advantageously, in many cases the uncertainty signaltypically has a smaller root-mean-square value and narrower effectivedynamic range than the signal representation. Also, because itemphasizes the unique and uncertain aspects of data, fewer resourcesneed be directed to processing the commonplace or expectable components.

For example, chromatic information in a color image may be subsampled toa greater extent without significantly noticeable degradation when it isfirst represented in terms of uncertainty, as opposed to theconventional representation as a linear combination of red, green, andblue intensity values. In addition, the inventive method of representinguncertainty does not require specific ad hoc assumptions about thecharacteristics of the input data. Thus, processing operations based onthe uncertainty signal will tend not to introduce errors resulting frominappropriate presumptions. The fact that the uncertainty signal has acompact, predictable dynamic range and distribution of values means thatit may be quantized more efficiently than is typically possible forsignal estimates or representations. Indeed, the quantization methoddescribed herein provides a means of representing the informativecontent of a signal estimate in terms of the quantized uncertaintysignal with minimal error and relatively few quantization levelscompared to typical histogram methods.

Because the uncertainty signal tends to preferentially representfeatures that are implicitly unexpectable, it can be used to emphasizeor de-emphasize features using simple arithmetic techniques without aneed to decide before hand which features may or may not be important.The same characteristic allows features to be extracted from data, orthe uncertainty signal itself, using simple threshold comparisontechniques. For example, edges, contrast discontinuities, and morecomplicated features such as the eyes of a face can be extracted fromimage data without having to define what constitutes an edge or eye byapplying a threshold comparison process to an image's uncertaintysignal. Alone and in combination, the inventive techniques allow data tobe categorized, identified, manipulated, compressed, coded, transmitted,and processed to achieve typical signal and information process goals insimpler ways than conventional methods and with minimal error orinformation loss. In addition, these techniques provide for new ways tocontrol and monitor processing operations.

FIG. 4 is a block diagram showing a signal representation S' operated onby an uncertainty task or bias G_(u), (box 400) to generate theuncertainty signal D' according to the inventive signal processingmethod, subjected to further processing steps (box 402), and thenoperated on by the inverse of the task G_(u) (box 404) to obtain a newestimate of the signal, S"(v). As shown in the figure, the uncertaintysignal D' is subjected to further processing steps to obtain a processeduncertainty signal, D"(v). This result is then operated on by theinverse of the task G_(u) (represented as 1/G_(u)) to obtain a newsignal representation, S"(v).

Processing operations suitable for implementation in box 402 include,for example and without limitation: quantization, de-quantization,subsampling and other means of resolution reduction, including any formof dithering; upsampling and other means of increasing apparentresolution including interpolation; DCT, FFT, and similar operations inwhich data are transformed to or from a frequency domain representation;wavelet-based and other convolution processes; fractal-type methods;coding and decoding methods including PCM, run-length methods, Huffmancoding, arithmetic coding, Lempel-Ziv-type methods, and Q-coding; andany combination of such operations or methods. Suitable processingoperations also include: permanent and/or temporary data storage;retrieval from stored sources; transmission; and reception.

The advantages in using an uncertainty signal in place of a signalrepresentation in processing operations are related to the uncertaintysignal's lower power, more compact and predictable distribution ofvalues, and tendency to preferentially represent the implicitlyunexpectable aspects of data. For example, in image processingapplications, for a given root-mean-squared difference between S' andS", the uncertainty signal can be quantized more coarsely and subsampledto a greater extent than S'. Similarly, in sampling processes, theamplitude of the uncertainty signal can be used to modulate the samplingrate or density in a linear, exponential, logarithmic, titration-like,or similar manner. The amplitudes and correlations in the uncertaintysignal may also be used as a guide for the positioning of basisfunctions. In addition, the absolute value, for example, of theuncertainty signal, rather than, or in conjunction with, the coefficientvalues of basis functions, may be used to control the number and/orvalues of basis-function coefficients that will be preserved in acompression process. Coding methods can be better tuned to data like theuncertainty signal which has a predictable distribution of values. Inaddition, the amount of power needed to transmit the uncertainty signalis less than would be needed to transmit S'.

Note that the present invention provides a signal processing method thatis not limited to linear operations having particular phasecharacteristics employed to estimate signal and/or noise from inputdata. By making the minimum number of assumptions regarding theattributes of the input data, applicants have been able to investigatehow conventional processing operations impose a signal and noisedefinition on input data. In some sense, every processing operation maybe viewed as a signal estimation process in which the result of theprocess represents the significant information content of the input datafor the particular application, as biased by the processing operation.The inventive method has clarified how the ambiguity of the resultingassignment of input data to either signal or noise should be representedgiven the assumptions implicit in the process.

An advantage and unusual feature of the inventive method is that it doesnot require any preconceptions with regard to what kind of signal andnoise model is implicit in a process. For a given process, it ispossible to interpret it in terms of any number of signal and noisemodels, regardless of whatever signal and/or noise characteristics theoriginal designer of the process may have had in mind. The fact that thepresent invention does not require an explicitly defined signal andnoise models means that it is versatile and robust.

However, applicants recognize that there are situations were there maybe a desire to use the inventive method to compare different processes,input data, signal representations, or uncertainty signals, as examples.In such cases, it would be beneficial to have a method in which implicitsignal and noise models could be judged by the same criteria; i.e., ifthey could be assessed by a standard method of interpretation. Forreasons noted, the method should make as few assumptions as possible. Itshould also be robust in the sense of being applicable to all possibleinputs.

Essentially, applicants have recognized that it would be advantageous ifthe invention provided methods for signal and noise characterization;i.e., if it provided a means of determining the processing function W(v)or its equivalents based on information such as the input data and theresulting estimated signal, and if it also provided a means of defininga signal and noise model given a processing function W(v) or itsequivalents. This permits a concise representation of the "black box"signal processing operations which have been implemented by a particularsignal processing system in a form which is compatible with the signalprocessing methods of the present invention.

As noted, the ability to define the signal component is related to howconstrained the signal is known to be. In conventional signal processingmethods, such knowledge must be available before processing the inputdata. However, for many data sources, including image sources, the"signal" is too variable to be defined in a precise manner. In thesecases, assumptions of what constitutes the signal must be applied. Theseverity of the misinterpretations that can result depends on thevalidity of the assumptions. In contrast, the present invention examinesthe implications of a particular signal and noise model and uses thatinformation to more efficiently process the input data or control anaspect of the processing.

Signal estimation processing according to the present invention isintended to make as few assumptions as possible for the reasons noted,which means that preferably the processing method should embody onlythat which is robustly expectable. It also means that the processingmethod should be designed to operate on classes of signals rather thanthe specifics of any particular signal. This broadens the range ofsignals and signal classes to which the inventive method cansuccessfully be applied.

The power spectrum of any particular set of data may be written as:

    |X(v)|.sup.2 =<|X(v)|.sup.2 >+δ|X(v)|.sup.2,

where <|X(v)|² > is the ensemble-average power spectrum and δ|X(v)|² isthe deviation from the ensemble average for the particular data set.Also,

    |X.sub.S (v)|.sup.2 =<|X.sub.S (v)|.sup.2 >+δ|X.sub.S (v)|.sup.2

    and

    |X.sub.N (v)|.sup.2 =<|X.sub.N (v)|.sup.2 >+δ|X.sub.N (v)|.sup.2

denote "signal" and "noise" components. Note that in the aboveequations, the deviation terms may take on both positive and negativevalues as opposed to a true power spectrum that everywhere is positiveor zero. The ensemble-average power spectrum is an average over allpossible sets of the input data. It is an overall expectation ratherthan a description of any particular set of data.

Likewise, the observed variance of any data set may be considered to bethe sum of an expectable component and a deviation from that expectablecomponent:

    σ.sub.X.sup.2 =<σ.sub.X.sup.2 >+δσ.sub.X.sup.2.

    Also,

    σ.sub.S.sup.2 =<σ.sub.S.sup.2 >+δσ.sub.S.sup.2 and σ.sub.N.sup.2 =<σ.sub.N.sup.2 >+δσ.sub.N.sup.2.

The ensemble-average variances are theoretical expectation values,whereas the deviations report the difference between the theoreticalvalue and the actual value for any particular set of data. A Poissonprocess, for example, has a theoretical variance equal to the meanintensity of the process, but actual observed variances will differ fromone observation period to another even if the mean intensity remains thesame.

The relationship between the ensemble-average power spectra and theensemble-average variances may be written as: ##EQU3## with similarequations for the deviation terms. The functions |K_(X) (v)|², |K_(S)(v)|², and |K_(N) (v)|² provide descriptions of the ensemble-averagepower spectra that are independent from variance. They are normalizedfunctions so that the integrated value over all frequencies of eitherfunction is identically 1, e.g., ##EQU4##

There are two forms of randomness that are generally associated withinput data: (1) the randomness of any noise disturbances that arerepresented by |X_(N) (v)|² and related terms; and (2) the randomness ofdeviations from expectations that are represented by terms such asδ|X(v)|². The deviation terms reflect ensemble variability. They areusually ignored because either the signal is considered to be completelyknowable a priori, in which case δ|X(v)|² =0, or the deviations are toounpredictable to be defined a priori.

In the classic estimation problem the goal is to produce a best guess asto the signal component of noisy data. Naturally, the guess must bebased on what is expectable and not on what are unpredictabledeviations. Except in cases where it is desired to give preferentialtreatment to particular subclasses of all possible stimuli (e.g., facesor square pulses), there is no real expectation that the signal and/ornoise components will have particular phase characteristics. Hence, theleast presumptive guess is based on expectations concerning powerspectra (or related functions such as correlation functions) alone;i.e., that the "signal" and "noise" components are not assumed to haveany expectable correlation, but rather it will be assumed that signaland noise are not correlated to some extent in a particular input. Inthe present invention, it is presumed that any signal and noisecorrelation in a particular input is not predictive of the signal andnoise correlations in all possible inputs. Thus, the present inventiondoes not presume any particular kind of signal and noise correlation. Insuch a situation the estimation processing function has an amplitudespectra of the form: ##EQU5## Processing operations having this generalform can be used to produce an estimate of a signal corrupted by noisewhere the signal and noise have objective definitions independent fromthe processing method. When discussing data representative of visualinformation, the inventors term such processes attribution processesbecause the ensemble-average signal correlations are really the resultof an imaging process, rather than statistically stable correlations inthe sources of visual data. Thus, although the form of the filter ismathematically similar to that of a Wiener filter, the assumptionsunderlying the use of such a filter function in the case of a signal andnoise model do not apply in the present situation. In signal processingaccording to the present invention, a signal and noise model may beassumed, however the invention is directed to an evaluation or analysisof the errors that can be introduced by that model.

As known and used, a Wiener filtering process requires that the signaland noise characteristics be defined and set a priori. The Wiener filterprocess would be judged to be appropriate only when the input wascomprised entirely of a signal and a noise having those predefinedcharacteristics. Any deviations from those characteristics would causethe implemented Wiener filter process to be suboptimal. For thesereasons, Kalman-type filters and other filter types which are capable ofadapting to changes in the input have largely replaced Wiener filteringprocesses. The mathematical form of a Wiener filter appears here in theexplication of the inventive signal processing method because it servesas a reference by which the least presumptive signal and noise modelimplicit in a processing operation may be characterized. As such, italso serves as a standard by which to interpret the inventiveuncertainty process, task, and signal.

The processing function expressed above weights input data according topower spectral density (the power spectrum evaluated at a particularfrequency). Frequency components in the input data that are more likelyattributable to the signal component than to the noise component (whenconsidered in terms of power density) are attenuated less than thosethat are more likely attributable to the noise component. Theattribution operation is thus graded in terms of relative expectablepower density.

In general, the estimation processing function may be written as:

    |W(v)|=[1+b.sup.2 B.sup.2 (v)].sup.-1

    where

    B.sup.2 =<|K.sub.N (v).sup.2 |>/<|K.sub.S (v).sup.2 |> and b.sup.2 =<σ.sub.N.sup.2 >/ >σ.sub.S.sup.2 <.

Any method of obtaining the appropriate combination of |X(v)|², |X_(N)(v)|², <σ² _(X) >, <σ² _(S) >, <σ² _(N) >, <|K_(S) (v)|² |>, <|K_(N)(v)|² >, b², or B² may be used to provide the terms needed to form|W(v)|. This includes user or external input, retrieval from a storagesource, averaging to obtain approximations, and input-output analysis ofexisting or hypothetical processing operations. Similarly, any means ofobtaining or defining |W(v)|, |U(v)|, or |G(v)| maybe used to providethe information required to characterize b² B² (v).

Note that b² B² (v) serves as the least-presumptive characterization ofa signal and noise model that is implicitly embodied in a processingoperation. It also serves as the least-presumptive signal and noisemodel that should be used in the inventive method.

With the signal estimation processing functions given above, theuncertainty processing function takes the form: ##EQU6## or

    |U(v)|=bB(v)[1+b.sup.2 B.sup.2 (v)].sup.-1

or an equivalent form.

Similarly, the uncertainty task is characterized by: ##EQU7## or

    |G(v)|=bB(v)

or an equivalent form.

Note that variance of the uncertainty signal is an indicator of theroot-mean-square error that can be expected in the estimation process.

The error in the signal estimation process can be written as: ##EQU8##

If the overall error in the signal estimation processes is written as

    ζ.sup.2 =<ζ.sup.2 >+δζ.sup.2

then it can be shown that <σ_(D) ² >=<ζ² >

Application of the Inventive Method to Processing Visual Image Data

Visual image data is a type of data particularly well-suited to beingprocessed using the inventive method. As described, there are twofundamental characteristics of visual information that createdifficulties for conventional processing methods. First, visualinformation is practically unconstrained. Visual data is any temporalseries, spatial pattern, or spatio-temporal sequence that can be formedby light. Whereas many signal processing problems make use of predefinedsignal characteristics (e.g., a carrier frequency, the transmitted pulsein a radar system, an alphabet), in many cases of interest, visualinformation arises from sources which are neither controlled norpredefined in any particular detail. Second, the very nature of lightitself creates ambiguity. Visual data can only be recorded as a seriesof photon-induced events, and these events are only statisticallyrelated to common parameters such as light intensity and reflectance.

The present invention provides several significant benefits whenprocessing such data:

(1) errors that can be expected in visual processing are reduced;

(2) important aspects of the data can be represented perceptuallywithout the imposition of ad hoc assumptions;

(3) visual information can be represented in a concise form having anarrow dynamic range and stable statistics;

(4) signals suitable for adaptation and error control can be produced;

(5) relatively simple devices can be used to implement the invention,thereby potentially reducing production costs; and

(6) the invention can produce indications of ambiguity, frequencycontent, and motion.

For any collection of objects distributed in space in any arrangement,essentially the only certainty is that the images of the objects will beof different sizes at the image plane. The associated power spectra sumlinearly because imaging is a linear phenomenon, and the compositespectrum will tend to fall off with frequency because the more distantobjects contribute less to the low frequencies than nearer objects. Whenintegrated over all possible arrangements of all possible objects, it isfound that the ensemble-average composite power spectrum tends to falloff with the inverse of the squared-value of the frequency coordinate.Such power spectra are called scale-invariant power spectra. Thearguments described above for spatial correlations are easily modifiedfor relative motion, leading to scale-invariance in the temporal domainas well.

Studies of the statistical characteristics of images have been reportedby: Field, D. J. (1987) Relationship between the statistical propertiesof natural images and the response properties of cortical cells. J. Opt.Soc. Am. A. Vol. 4:2379-2394; and Dong, D. W., Atick, J. J. (1995)Statistics of natural time-varying images. Computation in NeuralSystems. Vol. 6:345-358. These studies focused on the characteristics ofnaturally occurring images and image sequences. They found that amajority of individual natural images have an approximately 1/frequencyamplitude spectrum. However, applicants herein have found that manygraphic images and images of man-made objects do not have the1/frequency characteristic. However, to promote robust processing theinventive method described herein is directed to classes of inputsrather than to the particulars of individual inputs. Applicants havediscovered that, as a class, the ensemble-average amplitude spectrum ofimages has the 1/frequency characteristic.

Moreover, for subclasses of images, such as images of man-made objects,for example, the ensemble-average amplitude spectrum for the subclassalso has a 1/frequency characteristic, even though individual imagesvary significantly from the ensemble-average. The applicability of the1/frequency characteristic to images as a class may be considered to bea result of the process of forming an image.

In a general sense, visual images are the two-dimensional (2-D)accumulation of light from a three-dimensional (3-D) environment. Theobjects in the environment itself have no predictable orensemble-average statistical relationship to one another, but the act ofprojection introduces predictability, i.e., distant objects correspondto smaller images and take longer to transit a detector than do nearerobjects. This integration, resulting from the compression of 3-D depthinto a 2-D image, is described in the frequency domain by a 1/frequency²power spectra. Using the model of a 1/frequency² power spectra forvisual images, the term B² (v) in the inventive model is set equal tov². For the special case b² set equal to one, the processing functiontakes the form:

    W(v)=1/(1+v.sup.2).

Note that this frequency dependence of the above attribution processcharacterization is appropriate for any signal and noise model of theform A+v², where A is a constant.

The uncertainty filter, U(v) takes the form:

    U(v)=v/(1+v.sup.2).

A more general representation appropriate for any signal and noise modelof the form A+v² where A is a constant may be written as:

    W(v)=W.sub.0 [α.sup.2 /(α.sup.2 +v.sup.2)]

where W₀ is a scaling factor having a value of 1 when A=0. The parameterα² is related to b² and determines the frequency at which W(v) hashalf-maximal amplitude.

The corresponding uncertainty process function is:

    U(v)=W.sub.0 [α/(α.sup.2 +v.sup.2)][α.sup.2 (1-W.sub.0)+v.sup.2 ].sup.1/2

Those skilled in the art will appreciate from the within descriptions ofthe present invention the corresponding functional characterization ofthe uncertainty task.

For cases in which the randomness of photon capture is of primaryconcern, or in any case in which input data is representative of aPoisson process, α² may be taken to be a linear function of lightintensity (the mean rate of events in a Poisson process). For cases inwhich a fixed noise level is of primary concern, transducer or sensornoise, for example, α² may be taken to be a function of the square oflight intensity. In general, the value of α² may be determined bycomparing an equivalent of the r.m.s. (root-mean-square) power of anuncertainty signal to an equivalent of the r.m.s. power of an input.Note that the inventive method may also be extended to cases in which anoise of concern has an expectable power spectrum inversely proportionalto frequency. This sort of noise is often observed in electronamplifiers.

Hardware implementations of the above inventive process functions may bein the form of circuitry for real time processing. A minimum-phaseattribution process may be implemented as two identical stages ofsinge-pole low pass filters. The uncertainty process may implemented ina similar manner and is particularly straightforward when W₀ =1.

For spatial data, the attribution process may be implemented as atwo-dimensional equivalent of a transmission line in which a isrepresentative of a radial length constant. As such it may beincorporated into a sensor or implemented separately. A two-dimensionaltransmission line equivalent may be implement as a mesh of resistiveelements. Nodes in the mesh should have a resistive path to a commonground plane. The effective radial length constant of such animplementation may be controlled by modifying the resistance within themesh or the resistance in the ground path or both. Resistancemodifications may be achieved by using field-effect transistors orsimilar devices in a mode consistent with a voltage-controlled resistor.

Implementation in programmable devices may take the form of determiningdigital filtering coefficients consistent with the inventive method.Alternatively, data arrays equivalent to FFT representations may beconstructed and used in arithmetic combinations to process data.Functional descriptions and/or the equivalents of inverse FFTrepresentations may also be used in convolution operations. In digitalcomputing devices, it is some times computationally efficient andadvantageous to approximate a v-¹ as the reciprocal of the square rootof the absolute value of the FFT of an integer-valued array. A usefularray for one-dimensional data is [. . . -1 2 -1. . . ] where theellipses denote any number of zero-valued entries. A useful set ofarrays for two-dimensional data are of the form: ##EQU9## Where a>0 andthe surround of ellipses denotes any number of zero-valued entries. Forboth the one- and two-dimensional cases, arrays of non-zero valueslarger than 1-by-3 and 3-by-3, respectively, may also be used.Approximations of the type described may also be used to generate anarray of values for use in an equivalent of a convolution operation;e.g., by means of an inverse FFT. Computational efficiency inconvolution operations may be enhanced without introducing significantprocessing error by quantizing the values and/or limiting the number ofnon-zero elements in a convolution array.

In processing visual image data, the relationship between theuncertainty signal and the phase characteristics of the inputdemonstrates an advantage of the present invention. The uncertaintysignal essentially preserves the information in the phase of the inputand extends its utility beyond simple signal estimation. Any lineartransformation of the signal component can be estimated by passing theuncertainty signal through an appropriate linear filter or equivalent.Any nonlinear transformation may be produced from the uncertainty signalwith exactly the same quality as it could be produced from the signalestimate. Thus, the uncertainty signal does not exclude or restrictsubsequent processing operations. Instead, the uncertainty signal servesas a useful core signal on which any number of specific operations canbe performed in parallel. In accordance with the present invention, theuncertainty signal, instead of the traditional source intensity, may beconsidered as the primary signal in visual processing.

Most of the unique features of a particular set of visual data arerepresented by its phase spectrum. The amplitude spectrum describesoverall correlations without regard to when or where they occur. Thephase spectrum describes the locations and times of particular featureswithout regard to overall correlations.

The overall correlations in visual data have been described in terms ofexpectable power spectra. The unique details of any particular data setare therefore entirely described by the phase spectrum and the deviationof the particular power spectrum from the ensemble-average. These arethe components which contribute to the uncertainty signal. In a sense,the uncertainty signal for visual or any other data is representative ofan estimate of the phase characteristics of an input. However, unlike atrue whitening processes, the uncertainty process does not produce atrue representation of phase characteristics because the uncertaintysignal also represents aspects of an input which are indicative ofensemble variability.

One advantage of the present invention in visual processing is that theuncertainty signal emphasizes those aspects of the input data that aremost likely to distinguish that particular data set from data sets ingeneral. This is a perceptual emphasis as the unique features are thoseto which the human visual system is most sensitive. In essence, theuncertainty signal emphasizes the details of the input data and in oneaspect, the present invention may thus be considered a method of detailenhancement.

It is the lack of constraint of visual data that has been so problematicin other visual processing methods. Ad hoc assumptions concerningbiological visual performance have had to be made regarding what is andwhat is not perceptually important. The inventive method emphasizesfeatures without employing such assumptions, and hence is not prone toany of the disadvantages or bias effects resulting from suchassumptions.

Because the uncertainty signal preferentially represents details, it maybe used to enhance the perceptual qualities of the estimated signalcomponent. Note that the power spectrum of the expectable component ofthe estimated signal may be written as:

    <|X.sub.S.sup.40 (v)|.sup.2 >=|W(v)|)<|X.sub.S (v)|.sup.2 >.

The expectable component of the ambiguous power may be written as:

    <|D(v)|.sup.2 >=(1-|W(v)|)<|X.sub.S (v)|.sup.2 >.

Hence, their sum may be written as:

    <|X.sub.S.sup.40 (v)|.sup.2 >=<|D(v)|.sup.2 >=<|X.sub.S (v)|.sup.2 >.

This is another way of saying that the power in the uncertainty signalprovides a measure of the expectable error in a signal estimationprocess.

A further advantage of the present invention for visual processing isthat the uncertainty signal provides a means of boosting the frequencycontent of the signal estimate. Adding the uncertainty signal to theestimate of the signal component tends to sharpen perceptuallysignificant features such as edges and areas of sharp contrastdiscontinuities. Subtraction has the opposite effect, tending to blurthose features. The subtractive technique is useful in de-emphasizingthe pixelation apparent in low resolution images. The additive techniqueis useful in sharpening blurred text and aesthetic manipulation offaces, for example. The ease with which such image processing operationsmay be implemented using the methods of the present invention is asignificant benefit of the invention. Usually, such operations requirethe use of bandpass, highpass, and lowpass filters or equivalents ratherthan simple and efficient addition and subtraction, as is made possibleby the present invention.

Data Quantization

Just as the present invention provides a technique for reducing theprocessing errors introduced by adoption of a conventional signal andnoise model, it can also be used to develop a more efficient method ofquantizing data. Consideration of the same principles underlying thedata processing methods of the present invention permits development ofa quantization scheme which overcomes many of the disadvantages ofconventional methods.

The term quantization is used herein to mean the process by which theintrinsically continuous uncertainty signal is converted into a discretesignal. It is essentially an analog-to-digital conversion but thediscrete output need not be converted to binary form. The inventivequantization method is similar in concept to the attribution methodpreviously described. It produces a discrete version of the uncertaintysignal so that the statistically expectable difference from the originalis a minimum, thereby providing a quantization procedure which isconsistent with the fundamental assumptions of the invention

Applicants' quantization method described herein is not limited to datarepresentative of visual sources, and may be used to quantize datahaving any distribution of values. The quantization may be fixed in thesense of having predefined quantization levels, but the method describedcan also be used to adapt to changes in a distribution of values overtime. For visual data, it is often advantageous to expect that theuncertainty signal will have a Laplacian probability distribution (inthe ensemble-average sense) and to set the quantization levels accordingto that expectation. The quantization method may also be usediteratively; i.e., original data may be quantized, the quantizedrepresentation may then be compared to the original data or an updatedset of data, the difference between the quantized data and the referencedata may then be quantized. Source data may be approximated by summingthe successive iterations of quantization. This procedure is useful forspatio-temporal data such as video.

There are three sets of parameters required to understand the inventivequantization method: (1) state boundaries, (2) state numbers, and (3)state values, which can be referred to as interpretation values.Sequential pairs of state boundaries define the edges of a bin. Allvalues within the bin are assigned a state number. The state numbersform an integer series having N members, where N is the total number ofstates. The process of "binning" the uncertainty signal results in adiscrete version having N possible states. There are also N statevalues, but they do not necessarily form an integer series. Instead,they are determined so that the overall error in quantization isminimized. The state numbers are an index to the state values andboundaries.

The expectable integral squared quantization error (<ζ_(Q) ² >) may bewritten as: ##EQU10## Here δ_(n) are state boundaries, δ_(n) are statevalues, and n are the state numbers.

The integration parameter d represents the domain of the uncertaintysignal, not the actual values of a particular uncertainty signal. Thefunction ρ(d) is used to represent a histogram or probabilitydistribution.

The goal is to minimize the expectable error. Let Δζ_(n) ² denote theportion of the total error that is associated with state n; i.e.##EQU11## There are then two tasks: (1) find the set of state boundarieswhich minimizes error, and (2) find the set of state values thatminimizes error.

The selection of a state boundary influences the error associated withboth of the adjacent bins (states). Thus, it is necessary to find δ_(n)such that Δζ_(n) ² +Δζ_(n-1) ² is a minimum. The solution obtained viadifferentiation is: δ_(n) =(δ_(n) +δ_(n+1))/2, i.e., the state boundaryis exactly halfway between the state values. Thus, the state boundariesare completely determined by the state values.

The state value influences only the error associated with its own state.Using differentiation, the appropriate state value is equal to theintegral over the bin of dp(d) divided by the integral over the bin ofp(d). For a Laplacian distribution, the state values are given by:##EQU12## where β is the mean absolute value of d. For any particularuncertainty signal, β² =σ_(D) ² /2. The state values are best obtainedby noting that δ_(N) =(β+δ_(N-1)) because δ_(N) →∞. This gives astarting point from which to calculate other state values and stateboundaries using numerical methods or resistive ladders.

Note that exponential functions, like the Laplacian, display a sort ofscale-invariance. The shape of the function from any point, δ_(n), to ∞is exactly the same as from zero to ∞, the difference being simply inthe amplitude. With regard to state values, this means that the sequence(δ_(n) -δ_(n-1)) is independent of the total number of states. Thenumber of states dictates the number of elements of the sequence thatare relevant. In essence, increasing the number of quantization statesadds the new states near zero, thereby pushing the other stateparameters away from zero without changing their relationship to oneanother. Hence, (δ_(n) -δ_(n-1)) is a mathematical sequence that onlyneeds to be calculated once and stored; it need not be recalculatedevery time a signal is to be quantized. It is sometimes advantageous touse the recursive properties to quantize only the tails or otherportions of a distribution. Applications where this may be usefulinclude feature extraction, compression, and emphasis/de-emphasisoperations.

The usual means by which visual data is made discrete involve A-to-Dconversion of the intensity. For good quality images, the number ofstates employed is often 256 or greater. An advantage of the inventivemethod of quantizing the uncertainty signal instead of the input data isthat the same degree of quality as judged in terms of root-mean-squarederror is obtained with significantly fewer states (8 to 16 is typical).This significantly reduces storage capacity requirements.

Another advantage of the inventive quantization method stems from thefact that perceptually relevant aspects of the input data tend to beassociated with large values in the uncertainty signal. The quantizedversion may therefore be sorted by state value so that the informationmay be stored or transmitted in order of likely perceptual significance.This has implications for efficient image recognition, and the storage,transmission, and manipulation of a minimal set of data. The prior artmanner in which visual data is traditionally recorded in discrete formdoes not permit such a benefit.

Note that the error in quantization (the difference between the originaland the quantized version) also tends to have a Laplacian distribution.This means that the if the method is used recursively on a storedversion of the input data, or on a spatial array of input data thatvaries with time, it will continually update the quality of thequantized information without any additional effort or constraint.

Further with respect to the quantization process, conversion of theuncertainty signal into a discrete version of state numbers isindependent of the interpretation of those state numbers with statevalues. This means that the state number representation may be stored ortransmitted instead of the state value representation; i.e., the dynamicrange requirements are set by the number of states and not the power orrange in either the original uncertainty signal or input data. Thereceiver of the state number representation needs only to apply thealready known state values to obtain a minimal-error version of theoriginal uncertainty signal.

Efficiency of the quantization according to the present invention can beimproved if the uncertainty signal or other input is normalized by anestimate of its variance before being quantized. This allows theinterpretation values to be scaled as a group rather than individually.It also tends to reduce the "search time" when the state boundaries arefree to adapt to changes in the input. For data expected to have aLaplacian distribution, the variance of the data may be estimated fromthe mean absolute value of the data, thereby avoiding computationallymore intensity squaring operations.

General Applications of the Invention

Although the preceding exemplary description has emphasized applicationof the present invention for processing visual data, the invention maybe described as having three primary classes of applications:

(1) To generate a figure of merit to evaluate and permit comparisonbetween the effectiveness of different signal processing schemes;

(2) To generate a control term for use in adapting, modifying, or otherwise controlling the implementation of a signal processing operation;and

(3) As an intermediate form of processed data, to which other signalprocessing operations can be applied to perform further analysis in amore computationally efficient manner with reduced data storagerequirements. This form of using the invention facilitates datatransmission and compression operations, among others.

FIG. 5 is a block diagram depicting use of the present invention togenerate a figure of merit for purposes of monitoring a signalprocessing operation. As shown in FIG. 5a, in such an application of theinvention, the signal processing operations performed on a set of inputdata, X(v), to produce an estimated signal, S'(v) is characterized by a"black box" (labeled "Processing" in the figure). Both the input dataand estimated signal are represented as functions or data sets in ageneralized frequency space.

In this embodiment of the invention, input data, X, is operated on bythe uncertainty process U to produce the uncertainty signal D', whichmay then be input to one or more process monitors. Alternatively, D' maybe obtained from a signal estimate or representation, S', operated uponby G_(u), the uncertainty task, as shown in FIG. 5b. The signal estimateor representation may exist alone or be produced by operating on theinput, X, with an attribution process, W.

Process monitoring operations may include: comparing valuesrepresentative of D', such as the absolute value, quantized value,cumulative value, or root-mean-square power of D', to a set of definedvalues or functions; comparing values representative of transforms ofD', such as an FFT transform, to a set of defined values or functions;comparing data representative of variations in D' to a defined set offunctions such as a set of wavelet functions or other basis functions;producing a record, indicator, or alarm when certain relationshipsbetween D' and defined values and functions are met.

FIG. 6 is a block diagram illustrating how the inventive uncertaintysignal may be used as an intermediate form of processed data to replacea signal representation for the application of additional processingoperations. As shown in FIG. 6a, input data, X, which is typicallyprovided to a process (labeled "Processing" in the figure) is insteadoperated on by the uncertainty process, U, to produce D', theuncertainty signal. The uncertainty signal is then input to one or moreprocessing task operations (labeled "Tasks" in the figure).Alternatively, as shown in FIG. 6b, D' is obtained from a signalestimate, S', which is then operated on by G_(u), the uncertainty task.The signal estimate or representation may exist alone or be produced byoperating on the input, X, with an attribution process, W.

In addition to those possible tasks described in conjunction with FIG.5, other processing tasks can include, without limitation: thresholdingoperations in which only values of D' within a certain range are passedon to an output; translation and rotation operations; morphologicaltransformations such as warping or lensing distortions applied to imagedata for aesthetic effect; feature extraction using methods such asquantization, threshold, and frequency selection methods; featureemphasis and de-emphasis; root-mean-square normalization; andcombinations of such operations or methods.

FIG. 7 is a block diagram illustrating how the uncertainty signal may beused to control the operation of processes and/or processing tasks,according to the present invention. As shown in FIG. 7a, input data X,is subjected to a set of signal processing operations implemented by aprocessor (labeled "Processing" in the figure), and is operated upon bythe uncertainty process, U, to produce D', the uncertainty signal. Theuncertainty signal is provided to the processor as a control signal. Theprocessor may implement an attribution process, an attribution processin conjunction with one or more processing tasks, or may not bedivisible into separate attribution and task stages. Alternatively, asshown in FIG. 7b, the uncertainty signal may be obtained from a signalestimate S' which is then operated upon by G_(u), the uncertainty task.The signal estimate may exist alone or be produced by operating on theinput, X, with an attribution process, W. In any case, D' may optionallybe operated on by a control task, G_(c).

The uncertainty signal, or its post control-task representation, may beused to control: the selection of processes or processing tasks; therate at which data are to be sampled or coded; the amount by which dataare to be emphasized or de-emphasized; the selection of ditheringcharacteristics such as type, density, and diffusion; the dynamic rangeof a signal at any stage of processing by such means as variance orroot-mean-square power normalization; the amount and/or kind ofresolution reduction or enhancement; the number and/or kind orcoefficients to be retained in compression schemes such as JPEG, MPEG,fractal, and wavelet-based methods; the quantization criteria orthreshold levels to be applied to data; the characteristics of anattribution process; or any combination of such operations.

The control task operations can include, without limitation: means forproducing a signal that is representative of the root-mean-square valueof D' and/or S', and/or X'; rectification; quantization; thresholding;low-, band-, and high-pass filtering methods; and any combination ofsuch operations.

FIG. 8 is a block diagram illustrating a second manner in which theinventive uncertainty signal may be used to control the operation ofprocesses and/or processing tasks. The difference between FIGS. 8a and8b and FIGS. 7a and 7b is that in FIG. 8 the uncertainty signal is usedcontrol operations performed on the uncertainty signal.

FIG. 9 is a block diagram illustrating how the inventive signalprocessing methods may be used to perform data emphasis and de-emphasis.As shown in FIG. 9a, input data, typically a signal estimate or signalrepresentation, is presented to the uncertainty task, G_(u), which isthen scaled by a constant value, A. The result is added to the originalinput data to produce an output. FIG. 9b shows the same process buthaving the equivalent of two uncertainty tasks in series. The constant Amay be fixed, alternately, it may be controlled by a user or externalprocess.

The method shown in FIG. 9a preferentially adds or subtracts the uniqueand uncertain features of the input data to the input data therebyemphasizing or de-emphasizing those features. The method shown in FIG.9b provides a means of compensating for errors that may have beenintroduced during prior processing of the input data.

The value of the constant A may range from positive to negativeinfinity, although in practical applications values of A in the range ofplus and minus 1 will be sufficient. Positive values of A will produceemphasis, negative values will result in de-emphasis.

If applied to audio data, de-emphasis will tend to muffle sounds and/orreduce hiss, while emphasis will tend to have the opposite effect. Whenapplied to image data, de-emphasis will tend to be perceived as blurringor smoothing, whereas emphasis will be perceived as image sharpening andcontrast enhancement. Allowing (A) to be set by a user or externalprocess provides a means for controlling the dynamic range orroot-mean-square power of the output to achieve a desired perceptualcondition.

Typically, to achieve a continuous range of emphasis and de-emphasis, orsmoothing and sharpening, the properties of a filter or convolutionmethod need to be adjusted in a continuous manner. In essence, adifferent filter would be needed for each level of emphasis/de-emphasis.In contrast, the inventive method achieves a similar effect by adjustinga scalar multiplier.

FIG. 10 is a block diagram illustrating the use of the inventive signalprocessing method for constructing an uncertainty process from apre-existing or hypothetical signal or data processing operation(labeled "Process" in the figure). As shown in FIG. 10a, the Input andOutput of the Process are supplied to an input/output analysis block(I/O Analysis). The input is also operated upon by an uncertaintyprocess, U. Alternately, as shown in FIG. 10b, the Output may beoperated upon by an uncertainty task, G_(u), where the properties of theuncertainty process and/or uncertainty task are determined by theresults of the I/O Analysis. Typically, that actual processing of theinput data by the uncertainty process or task would be performed by aprogrammable device by convolution, digital filtering, or arithmeticoperations performed on frequency domain representations. Optionally, auser or external process such as a database system, may provide scalingand processing task information to the I/O analysis method.

Given that the output of a processing operating may depend in somenon-linear manner on the characteristics of the input data, the methoddescribed above provides an adaptive means of quantifying the ambiguityinherent in the relationship between the input, output and processingmethod, as well as a means of characterizing the processing method andthe associated ambiguity. This characteristic of the method may proveadvantageous when the input data is derived from several differentsources or prior processing operations, such as may be the case in amultiplexing system. Typically, a processing method needs to be designedand implemented to encompass the degree of freedom allowed to the rangeof possible inputs. In many cases, the range of inputs and their degreeof freedom has to be constrained to satisfy the need for processingefficiency. In contrast, an advantage of the inventive method is thatproviding the uncertainty signal for use in process monitoring andcontrol reduces the tightness of the constraints which might otherwisebe necessary in the design of inputs and processing operations.

FIGS. 11a and 11b are flow charts showing the primary signal processingsteps which are implemented to determine the uncertainty filter, U(v),and uncertainty task, G_(u), based on an I/O analysis of a processingscheme according to the method of the present invention. As shown inFIG. 11b, the I/O analysis described with reference to FIGS. 10a and 10bcan be used to provide the information required to construct theuncertainty task. This is both necessary and sufficient to construct anattribution process and an uncertainty process.

Information equivalent to a scaling constant, A, and an estimate of theamplitude spectrum of the effective input/output response function, |F|,is sufficient to define the uncertainty task. Optionally, informationequivalent to an amplitude-spectrum description of a known or supposedprocessing task, |G_(f) | may be supplied. If |G_(f) | is not available,it may be set to a value of 1 for all values of the generalizedfrequency.

Given the input data and estimated signal, the process function F(v) isdetermined from |F(v)|=|Y(v)|/|X(v)|, as shown in the figures. Next, theprocessing task function, G_(f) (v), is assumed, where F(v)=G_(f)(v)W(v), and W(v) is the generalized signal interpretation function. Asnoted, G_(f) (v) may be a smoothing operator, or other form of weightingfunction, with the constraint that 0≦|G_(f) (v)≦|F(v)| for all v. Next,the following term for the signal interpretation function is formed:

    |W(v)|=|F(v)|/(A|G.sub.f (v)|).

The scaling constant, A, is adjusted as needed to satisfy the conditionof max |W(v)|<1, by setting

    A=max (|F(v)|)/max (|G.sub.f (v)|).

The uncertainty process function |U(v)| function is then obtained from:

    |W(v)|(1-|W(v)|)).sup.1/2,

as shown in FIG. 11a. The uncertainty processing task, G_(u), may alsobe formed as a result of the I/O analysis from:

    ((1-|W(v)|)/|W(v)|).sup.1/2,

as shown in FIG. 11b.

Note that the expressions for |U(v)| and G_(u) (v) do not specify thephase characteristic of the respective processes. In cases where theinput and/or output data takes the form of an array, such as is the casefor a still image, and in cases where input and/or output data is storedin a buffer while awaiting processing, it is appropriate these functionshave a zero-phase characteristic. In cases in which it is desirable thatdata be processed in real time (or nearly so), it is preferable that thefunctions have phase characteristics which are as close as possible tothose which characterize the class of filters known as minimum-phasefilters. Implementation of such filters is known to those skilled in therelevant art, and can include methods related to spectrum and cepstrumanalysis.

Several approaches may be used in estimating |F|. Arguably the simplestapproach is to estimate |F| from spectral estimation of a stored exampleof the input and output data, or from averages derived from severalinstances of spectral estimation.

The I/O analysis described with reference to FIGS. 11a and 11b providesa representation of the signal-to-noise characterization inherent in theblack box of the signal processing operations. Under some circumstancesit may be more readily determined than a signal-to-noise ratio based onconventional definitions and processing methods.

Applying U(v) to the input data, X(v) provides the ambiguous component(previously termed D'(v)) of the processing relationship described byF(v). This is a figure of merit which indicates the quality of theprocessing operations used to extract the signal estimation from theinput data. A similar figure of merit may be determined for multiplepossible processing operations and compared to decide which suchoperation will process the input data while reducing the errors in theprocessing scheme arising from the imposed signal to noise model.

Another application of the present invention in image processing is topartition an image into a set of blocks and use the uncertaintyrepresentation to compare the benefit of each of a group of possibleimage processing operations on each block. This permits the selection ofthe "optimal" processing operation for each block, thereby providinganother method of enhancing or correcting image data.

FIG. 12 is a block diagram illustrating methods of implementing theattribution process, uncertainty process, uncertainty task, and relevantinverse processes in accordance with the present invention. Therelationship between the attribution process, uncertainty process, anduncertainty task provides a significant degree of flexibility in theprocessing scheme used to obtain the benefits of the present inventionbecause any one of the processes or tasks may be obtained using theother two and/or their inverses. Note that the order of the operationsshown in the figure is not the only one capable or providing the desiredend result. The sequence of operations shown are preferred for mostapplications but variations are also possible.

The present invention affords several advantages when implementing thesequence(s) of operations shown in the figure. When implemented by aprogrammable device, the process functions may be represented in formssuch as discrete frequency-domain representations, digital filtercoefficients, and/or convolution matrices. Less storage space isrequired to store two such representations than would be required tostore all three. In addition, implementation of one of the processes bymeans of the others will typically provide useful intermediate results.

For example, the use of U and G_(u) to obtain an attribution process, W,will produce D', the uncertainty signal, and N', a noise estimate, inaddition to S', the signal estimate. In this and similar cases, there isa saving of computational resources because the step of addition used toproduce S' is simple compared to a convolution operation or anequivalent, which would be required in many conventional signalprocessing schemes. In some cases there may also be a reduction in datastorage requirements because D', for example, contains informativecontent sufficient to produce S' and/or N'. Hence, in this case only D'would need to be stored for subsequent operations to produce S'. Theimplementation of one process by means of the other two also allows forother intermediate processes to be inserted or performed in parallel.For example, in the implementation of W by means of U and G_(u), theintermediate result D' may be subjected to another processing operationsuch as coding/decoding, resolution reduction, compression,quantization/dequantization, transmission/reception, storage/retrieval,or any combination of similar operations.

The intermediate result can also be extracted for use in processmonitoring and/or control. Another advantage of such a modular method ofimplementation is that it may be used to avoid problems with therealizability of any one of the processes. For example, it may not bepossible or perhaps is simply inefficient, to directly construct aparticular instance of an attribution process. Such a situation mayoccur when round-off errors and/or division-by-zero issues arise inprogrammable devices. In hardware implementations, it may provedifficult to implement a process having the appropriate responseprofiles in terms of both phase and amplitude, while another of theprocesses or their inverses may be more easily or efficientlyimplemented. In these cases, the modular method provides for"work-around" solutions.

As an example, the uncertainty process for data representative of atwo-dimensional image may not be efficiently implemented outright.However, the appropriate attribution filter may be constructed as theequivalent of a two-dimensional transmission line and may be built intoan image sensor. Hence, the uncertainty process can be performed byimplementing the attribution process followed by G_(u). G_(u) could beimplemented by any means which effectively resulted in spatialdifferentiation. Alternately, U could be obtained by mean of W and1/G_(u). For this example, G_(u) could be approximated by use of anothertwo-dimensional transmission line equivalent having a characteristicradial length constant at least several times larger than that of theattribution process, or it could be implemented by any other process,such as an accumulator, which would effectively result in spatialintegration.

FIG. 12a illustrates two representative methods of implementing anattribution process operating on an input S+N, to produce S'. In oneexample, U operates on the input to produce D' which is then operatedupon by 1/G_(u) to produce S'. In another example, U operates on theinput to produce D' which is then operated on by G_(u) to produce N',which is then subtracted from the input to produce S'.

FIG. 12b illustrates two representative methods of implementing anuncertainty process operating on an input, S+N, to produce D'. In oneexample, W operates on the input to produce S' which is then operated onby G_(u) to produce D'. In the other example, W operates on the input toproduce S' which is then subtracted from the input to produce N', whichis operated upon by 1/G_(u) to produce D'.

FIG. 12c illustrates two representative methods of implementing anuncertainty task operating on an input S', to produce D'. In oneexample, U operates on the input to produce US' which is then operatedon by 1/W to produce D'. In the other example, W operates on the inputto produce WS' which is then subtracted from the input to produce(1-W)S', which is operated upon by 1/U to produce D'.

FIG. 12d illustrates a method of implementing an inverse attributionprocess. Such a process may be used as a intermediate process asdescribed above. It may also be used to operate on a signal estimate orrepresentation, S', to provide an estimate of signal and noise, (S+N)'.In the example shown, the equivalent of two stages of the uncertaintytask, G_(u) operates on the input and the result is added to the input.

FIG. 12e illustrates a method of implementing an inverse uncertaintyprocess. Such a process may be used as a intermediate process asdescribed above. It may also be used to operate on an uncertaintysignal, D', to provide an estimate of signal and noise, (S +N)'. In theexample shown, 1/G_(u) and G_(u) operate upon the input in parallel andthe results are added.

Hardware and Software Implementations of the Embodiments of theInvention

The various embodiments of the inventive signal and image processingmethods disclosed herein may be implemented in several forms. Theseinclude: (1) programming of a digital computer to implement the methodsteps as software based on the flow charts and processes describedherein; (2) processing of input signals by circuitly of the typedisclosed in the copending provisional application; and (3) processingof input signals by dedicated processing structures.

In practice, a computer system having a pentium-class central processorunit (CPU) that executes one or more software routines, preferablystored or storable in memory associated with the computer system, issufficient to carry out the present invention. The CPU executes theroutine(s) embodying one or more of the methods described herein. Ifdesired, a general purpose programmable signal processor could be usedinstead of a computer system. Such signal processors are known to thoseskilled in the art and are commercially available from a number ofvendors, Texas Instruments, Inc. for example.

Some additional comments on various embodiments and implementations ofthe present invention may be useful.

Ensemble-Average Power Spectra

To produce data representative of an ensemble-average power spectrum,the following procedures are suggested:

(1) choose a class of inputs (still images, for example);

(2) record data representative of a member of the input class with anappropriate sensor;

(3) sample the output of the sensor;

(4) convert the sampled data using an analog-to-digital converter (ADC);

(5) store a specified number of samples;

(6) perform a fast Fourier transform (FFT) on the stored data;

(7) square the absolute value of the FFT data to obtain an estimate of apower spectrum;

(8) store the estimated power spectrum;

(9) repeat steps (2) through (7) for another representative of the inputclass, modify step (7) so that the new estimated power spectrum is addedto the data currently stored so that the stored data represents the sumof all estimated power spectra that have been computed;

(10) repeat the process until a desired number of members of the inputclass have been processed;

(11) divide the summed power spectra data by the number of iterations sothat the result is representative of an average, the result being anestimate of the ensemble-average power spectrum for input class. Thenumber of iterations required to obtain a reliable estimate of theensemble-average power spectrum will vary depending on the input class,but fewer than 20 iterations will typically sufficient. A estimatedpower spectrum may also be obtained by fitting curves, splines, and/oranalytic functions to the averaged power spectrum obtained by the stepslisted above. A power spectrum may be normalized by dividing each datapoint by the sum over all data points.

If the input data exist originally in digital form, steps (1) through(5) are not required.

Estimates of ensemble-average power spectra for noise components canoften be modeled based on knowledge of the nature of input data orknowledge of the characteristics of sensing devices, amplifiers, andother components. For example, the quantal randomness of photon capturecan be modeled as a white noise process even though it is a form ofintrinsic randomness. Most sensors have a thermal noise that can berecorded in the absence of an input signal to produce estimated powerspectra as noted, or modeled based on information supplied by themanufacturer. Typically, sensor and amplifier noise can be modeled as awhite noise process and/or a 1/f noise process. In the case where thereis no way of reliably determining or characterizing noise, the noisepreferably is modeled as white noise because there is no reason tosuppose that any particular frequency range contributes to uncertaintyany more than any other frequency range.

Processing Functions

In constructing digital representations of the processing functionsW(v), U(v), and/or Gu(v), it should be noted that B(v) will take theform of a linear array or matrix of elements. The term B² (v) isobtained by squaring each element of B(v); i.e., B(v) is multiplied byB(v) element by element. Division operations should also be performedelement-by-element. Similarly, an operation such as [1+B(v)] indicatesthat one should add 1 to each element of B(v).

As frequency domain representations, processing functions may bemultiplied by FFT-versions of input data to yield desired results.Alternatively, inverse FFT operations may be performed on the frequencydomain representations of the processing functions to yield arepresentation suitable for convolution operations.

Minimum-phase versions of processing functions may be obtained using thefollowing procedures:

(1) constructing the processing function without regard to phasecharacteristics;

(2) taking the absolute value of the processing function;

(3) performing an inverse FFT;

(4) using a function such as rceps() available from The Mathworks, Inc.which returns a minimum-phase version of the inverse FFT. Theminimum-phase result may be convolved with input data. Alternatively,one can calculate the FFT of the minimum-phase result to yield aminimum-phase frequency domain version of the processing function.

Data Processing

A preferred method of processing data according to the present inventionis as follows:

(1) record data representative of a member of the input class with asensor;

(2) sample the output of the sensor;

(3) convert the sampled data using an analog-to-digital converter (ADC);

(4) store a specified number of samples;

(5) perform a fast Fourier transform (FFT) on the stored data;

(6) multiply the FFT data, element by element, by a FFT-version(frequency domain representation) of a processing function;

(7) perform an inverse FFT on the result; and

(8) repeat the process as desired.

Equivalently, data may be processed using the inventive method byperforming steps (1) through (4)as above, and (5) convolving the storeddata with an appropriate representation of the processing function.

Adaptation

The term b² as described herein is an "optimization parameter"representative of a ratio of noise variance to signal variance. Thereare several methods by which its value may be set.

In some cases, it is advantageous to allow a user to set the value ofb². For example, a user may input a desired value to a computer programor control the value using a dial connected to a potentiometer. Such amethod may be suitable in cases where a user desires to control theperceptual aspect of image, video, or audio data, for example.

In cases in which it is known or assumed that the r.m.s. power of noiseis fixed or relatively constant, the value of b² may be estimated usingthe following procedures:

(1) calculating the r.m.s. value of the input data;

(2) squaring the r.m.s. input value to yield an estimate of the inputvariance;

(3) calculating the difference between the input variance and the known,estimated, or assumed noise variance to yield an estimate of the signalvariance;

(4) calculating the ratio of the noise variance to the difference ofvariances.

In cases where the noise variance is known or assumed to be small withrespect to the input variance, step (3) need not be performed and theinput variance may be taken as an estimate of the signal variance. Thoseskilled in the art will recognize that an equivalent procedure may beused if the variance of the presumed signal component or input varianceis known or expected to be fixed or relatively constant. Variances mayalso be estimated for digital data by determining the mean squared valueof the data.

In the case of data derived from light, it is known that the randomvariation due to photon capture contributes a variance to the input inproportion to the mean light intensity. The variance of the "signal"component increases as the square of the mean light intensity. Thus,allowing for dark noise in a light sensor, the value of b² may bedetermined from the mean light intensity rather than from inputvariances, for example. Where photon randomness is the predominantsource of "noise," the value of b² should be inversely proportional to alinear function of light intensity. Where other noise sources having afixed r.m.s. power dominate, b² should be inversely proportional to afunction of light intensity squared. The mean light intensity may beestimated by means of a low-pass filter connected to a light sensor, orby other means of averaging.

In other cases, the value of b² may be set by a method of minimizing ther.m.s. value of the uncertainty signal with respect to the r.m.s. valueof the input. One such method preferably carries out the followingsteps:

(1) recording and storing an input;

(2) selecting an initial value of b² ;

(3) processing the input by the inventive method to produce anuncertainty signal;

(4) forming and storing a ratio of the r.m.s. value of the uncertaintysignal to the r.m.s. value of the input;

(5) selecting a new value of b² ;

(6) producing a new uncertainty signal;

(7) forming a new ratio of r.m.s. values;

(8) comparing the first ratio to the second ratio; if the new value ofb² is greater than the first value and if the value of the second ratiois greater than that of the first ratio, then a new, lesser value of b²needs to be selected and the process repeated until a value of b² isfound such that any increase or decrease in its value results in agreater ratio of r.m.s. values. Those skilled in the art will recognizethat algorithms are known with which to search for a minimum value.

Data Manipulation

As noted, the present invention advantageously provides a means ofextracting features from data based on the value of an uncertaintysignal. For example, regions near the eyes, nose, mouth, hairline, andoutline of a face may be preferentially extracted from an image of aface by retaining values of an uncertainty signal which exceed a certainlimit. One method of achieving feature extraction preferably includesthe following steps:

(1) obtaining input data;

(2) producing an uncertainty signal;

(3) normalizing the uncertainty signal by its standard deviation;

(4) comparing the absolute value of the uncertainty signal to a setlevel;

(5) storing the value 1 at each point at which the threshold is exceededand the value of 0 wherever it is not.

A threshold value in the range of 1 to 3 works well for images of faces.The non-zero values in the resulting binary map tend to mark locationsof maximum ambiguity or uncertainty. For images, these areas tend to beperceptually significant and useful in recognition processes. The mapmay be multiplied by the input data or a signal representation so thatonly those areas of the input or signal data corresponding to a 1 in thebinary map are preserved. Alternatively, the binary map may bemultiplied by the uncertainty signal. The result may be processed by theinverse uncertainty task of the inventive method to produce arepresentation of a signal only in those areas corresponding the largemagnitude values of the uncertainty signal. An additional step ofquantizing the uncertainty signal may be included before or after thethreshold comparison.

The feature-extraction method may be used in conjunction withsubsampling/interpolation operations so that data corresponding tolarger values of the uncertainty signal are preferentially retained. Asan example, having obtained an uncertainty signal, the following stepspreferably are carried out:

(1) produce a binary map representing location at which the absolutevalue of the uncertainty signal exceeds a defined limit;

(2) multiply the binary map by the uncertainty signal and store theresult;

(3) subsample the uncertainty signal by averaging neighboring elementsso that the result has fewer elements than the original uncertaintysignal;

(4) interpolate the subsampled uncertainty signal so that the result hasthe same number of elements as the original;

(5) multiply the result by a binary map produced by performing a NOToperation on the original binary map;

(6) add the result to the stored product of the original binary map andthe original uncertainty signal.

Additionally, the result may be processed by an inverse uncertainty taskof the inventive method to produce a representation of a signal in whichthe details near locations of large uncertainty signal values arepreferentially preserved. The steps described may be used in a pyramidalmethod in which certain areas of an uncertainty signal are preserved ateach level of resolution.

A similar method of preferentially preserving resolution in certainareas involves adjusting a sampling rate or density in accordance withthe value of the uncertainty signal. For example, an absolute value ofan uncertainty signal may be used as a parameter in a linear functionwhich determines the inter-sample duration so that input, signal, oruncertainty data are sampled at the end of each duration period.Provided the sample duration decreases with increasing absolute value ofthe uncertainty signal, data will be sample at a higher rate nearlocations of large magnitude values of the uncertainty signal. Theduration period may be set by the value of the uncertainty signal at theend of the preceding interval, or by the average absolute value of theduring the preceding interval, for example. Methods of this kind mayalso preserve the sign of the uncertainty signal so that negative valuesand positive values do not have the same effect.

Another method of controlling resolution and quality, having obtained anuncertainty signal, preferably involves the following steps:

(1) determining the mean absolute value or variance of the uncertaintysignal with respect to a certain duration or area;

(2) setting an effective bandwidth as a function of the result of step1;

(3) processing data in accordance with the criterion of step (2) so thatonly a certain bandwidth of the processed data is preserved. As examplesof step (2), the mean absolute value may correspond in a linear mannerto the low-frequency cut-off of a high-pass filter, or it may correspondto the high-frequency cut-off of a low-pass filter. Equivalently, inbasis-function methods, such as JPEG, the uncertainty signal may be usedto control the number of coefficients to be preserved in a certainduration or area of processed data. In wavelet-based methods, the rangeof allowed scaling factors may be controlled.

To recapitulate, the present invention provides a method of analyzingand representing data which can be used to evaluate the ambiguity orerror introduced by a particular signal and noise model of the data.This permits computationally efficient representation and manipulationof data without the introduction of bias from assumptions as to thenature of the data or relationships between different pieces of data.The inventive method is of particular use in data compression andtransmission, as well as the processing of image data to emphasize orde-emphasize specific features.

The terms and expressions herein are used as terms of description andnot of limitation, and there is no intention in the use of such termsand expressions of excluding equivalents of the features shown anddescribed, or portions thereof, it being recognized that variousmodifications are possible within the scope of the invention claimed.Thus, modifications and variations may be made to the disclosedembodiments without departing from the subject and spirit of theinvention as defined by the following claims.

What is claimed is:
 1. A method of processing a set of input data X(v)representing a desired signal component plus an undesired contaminationcomponent, the method comprising the following steps:obtaining anensemble-averaged power spectrum of the signal component, <|K_(S)(v)|² >; obtaining an ensemble-averaged power spectrum of thecontamination component, <|K_(N) (v)|² >; forming a term B² =<|K_(N)(v)|² >/<|K_(S) (v)|² >; forming a filter function W(v), where|W(v)|=[1+b² B(v)² ]⁻¹, and b is greater than zero; forming a term U(v),where |U(v)|=[W(v)(1-W(v))]^(1/2) ; and processing X(v) to form a resultU(v) X(v).
 2. The method of claim 1, wherein the step of obtaining anensemble-averaged power spectrum of the signal component, furthercomprises:averaging a set of data known to represent the signalcomponent of the set of input data.
 3. The method of claim 1, whereinthe step of obtaining an ensemble-averaged power spectrum of thecontamination component, further comprises at least one step selectedfrom a group consisting of (i) assuming a model for the contaminationcomponent and determining its ensemble-averaged power spectrum, and (ii)assuming a model for the contamination component that is arepresentation of white noise.
 4. The method of claim 1, furthercomprising at least one step selected from a group consisting of (i)further processing the result U(v) X(v) by applying a desired signalprocessing technique, and (ii) quantizing the result U(v) X(v).
 5. Themethod of claim 1, further including the step of quantizing the resultU(v) X(v), wherein said quantizing includes at least one step selectedfrom a group consisting of (i) comparing the result to a set ofreference values determined from the result's ensemble averageprobability density function and generating a quantization value for theresult corresponding to a member of the set of reference values, and(ii) comparing the result to a set of reference values determined from amodel of an ensemble average probability density function and generatinga quantization value for the result corresponding to a member of the setof reference values.
 6. The method of claim 1, wherein b² is inverselyproportional to [a₁ +I^(n) ], where a₁ is a constant, I is a mean valueof the signal, and n is an integer.
 7. The method of claim 1, whereinthe set of input data is representative of visual image data.
 8. Amethod of processing data representative of visual images, the methodcomprising the following steps:forming a filter function W(v), where|W(v)|=[1+b² B(v)² ]⁻¹, b² is a constant selected to satisfy |W(v)|<1for all v, and B(v) is proportional to v; forming a term U(v), where|U(v)|=[W(v)(1-W(v))]^(1/2) ; and processing X(v) to form result U(v)X(v).
 9. The method of claim 8, further comprising at least one stepselected from a group consisting of (i) further processing the resultU(v) X(v) by applying a desired signal processing technique, and (ii)providing the result U(v) X(v) as a control term to vary amplitudespectrum of the function filter function W(v).
 10. The method of claim9, further comprising at least one step selected from a group consistingof (i) quantizing the result U(v) X(v), (ii) quantizing the result bycomparing the result to a set of reference values determined from theresult's ensemble average probability density function and generating aquantization value for the result corresponding to a member of the setof reference values, and (iii) quantizing the result U(v) X(v) bycomparing the result to a set of reference values determined from amodel of an ensemble average probability density function and generatinga quantization value for the result corresponding to a member of the setof reference values.
 11. The method of claim 9, wherein b² is inverselyproportional to [a₁ +I^(n) ], where a₁ is a constant, I is a mean valueof the signal, and n is an integer.
 12. A method of characterizing adata processing operation which processes input data X to form a resultY, comprising the following steps:forming a function W, where |W| isproportional to |Y|/|X|; forming a function U, where |U| is equal to

    [|W|(1-|W|)].sup.1/2 ; and

applying the function U to the input data X to obtain the result U(v)X(v).
 13. The method of claim 12, wherein input data X is representativeof visual image data.
 14. The method of claim 12, including a step offurther processing the result U(v) X(v) using at least one techniqueselected from a group consisting of (i) applying a desired signalprocessing technique, (ii) providing the result U(v) X(v) as a controlterm to vary amplitude spectrum of the function filter function W(v),(iii) quantizing the result U(v) X(v), (iv) quantizing the result U(v)X(v) by comparing the result to a set of reference values determinedfrom the result's ensemble average probability density function andgenerating a quantization value for the result corresponding to a memberof the set of reference values, and (v) quantizing the result U(v) X(v)by comparing the result to a set of reference values determined from amodel of an ensemble average probability density function and generatinga quantization value for the result corresponding to a member of the setof reference values.
 15. A method of characterizing a data processingoperation that processes input data X to form a result Y, comprising thefollowing steps:forming a function W, where |W| is proportional to|Y|/|X|; forming a function Z, where |Z| is equal to

    [(1-|W|)/|W|].sup.1/2 ; and

applying the function Z to the output data Y to obtain the result Z(v)Y(v).
 16. The method of claim 15, wherein input data X is representativeof visual image data.
 17. The method of claim 15, further comprising thestep of: further processing the result Z(v) Y(v) using at least one stepselected from a group consisting of (i) applying a desired signalprocessing technique to the result Z(v) Y(v), (ii) providing the resultZ(v) Y(v) as a control term to vary amplitude spectrum of the functionfilter function W(v), (iii) quantizing the result Z(v) Y(v), (iv)quantizing the result Z(v) Y(v) by comparing the result to a set ofreference values determined from the result's ensemble averageprobability density function and generating a quantization value for theresult corresponding to a member of the set of reference values, (iv)quantizing the result Z(v) Y(v) further by comparing the result to a setof reference values determined from a model of an ensemble averageprobability density function and generating a quantization value for theresult corresponding to a member of the set of reference values.
 18. Asignal processing system, comprising:a data input node for inputting asignal X to be processed; a digital computing apparatus including atleast a central processor unit and memory programmed to operate on aninput signal to implement operations of:forming a first processingfunction W, wherein the amplitude spectrum W(v) is given by

    W(v)=(1+b.sup.2 B(v).sup.2).sup.-1 ;

where b is a constant and B(v) is a positive, real valued function offrequency; forming a second processing function U having an amplitudespectrum given by

    [|W|(1-|W|)].sup.1/2 ;

applying the second processing function U to the input data; performinga desired signal processing operation on the result of applying thesecond processing function U to the input data; and a display device fordisplaying a result of operations implemented by said digital computingapparatus.
 19. The signal processing system of claim 18, wherein inputsignal X is representative of visual image data.
 20. The signalprocessing system of claim 18, wherein B(v) has at least onecharacteristic selected from a group consisting of (i) B(v) isproportional to v, (ii) B² (v) is proportional to [a₁ +v^(n) ]/[a₂+v^(m) ] where a₁ and a₂ are constants and n and m are integers, (iii)B² (v) is represented by vector [0 . . . 0 -1 2 -1 0 . . . 0] whereellipses represent any number of zeros, and (iv) where a>0 and B² (v) isrepresented by a matrix: ##EQU13##
 21. A method of processing a signal,comprising the following steps: providing a signal generated by astochastic process, the signal including a stochastic source componentand a random process component;inputting the signal to a filter havingan amplitude spectrum W₁ (v), where v is a frequency component of thesignal, the filter output representing a weighting of the frequencycomponent a greater amount if it is more likely attributable to thestochastic source than to the random process; and weighting the outputof the filter by a function having an amplitude spectrum W₂ (v), whereW₂ (v) is given by

    [(1-W.sub.1 (v))/W.sub.1 (v)].sup.1/2.


22. The method of claim 21, further comprising the step of:providing theweighted output of the filter as a control signal to vary the amplitudespectrum of the filter, wherein the control signal has an expectablepower spectrum proportional to |W₁ (v)W₂ (v)|.
 23. The method of claim21, further comprising the step of quantizing an output of the filter socontrolled.
 24. The method of claim 21, wherein amplitude spectrum W₁(v) is represented by W₁ (v)=[1+b² B(v)² ]⁻¹, where b is a constant andB(v) is a positive, real valued function of frequency.
 25. The method ofclaim 24, wherein B(v) is proportional to v.sup.±n, where n is aninteger.
 26. A signal processor, comprising:means for inputting dataX(v) representing a desired signal component plus an undesiredcontamination component; filter means for filtering the data so input,the filter means having an amplitude spectrum W(v), where |W(v)|=[1+b²B(v)² ]⁻¹, B(v) is a positive, real valued function, and b is a positivenumber; means for weighting an output of the filter means by U(v), where|U(v)|=[W(v)(1-W(v))]^(1/2) ; means for providing a weighted output ofthe filter means as a control signal to vary amplitude spectrum of thefilter means; and means for processing X(v) to form the result U(v)X(v).
 27. The signal processor of claim 26, wherein B(v) is proportionalto v.sup.±n, where n is an integer.
 28. The signal processor of claim26, wherein the filter means includes a resistive network having anadjustable parameter which is varied by a control signal.
 29. The signalprocessor of claim 28, wherein the adjustable parameter is a ratio ofsheet resistance to shunt resistance.
 30. A method of enhancing regionsof an image in which contrast discontinuities are present, the imageformed from a plurality of signals generated by a stochastic process,wherein each signal includes a stochastic source component and a randomprocess component, the method comprising the following steps:weightingthe signals by a filter function having an amplitude spectrum W₁ (v),where v is a frequency component of the signal, the amplitude spectrumacting to selectively weight the frequency component greater if it ismore likely attributable to the stochastic source than to the randomprocess; weighting the output of the filter function by a functionhaving an amplitude spectrum W2(v), where W₂ (v) is given by

    [(1-W1(v))/W.sub.1 (v)].sup.1/2 ; and

adding a result of weighting output of the filter function by W₂(v) tooutput of the filter function to form components of an enhanced image.31. The method of claim 30, wherein amplitude spectrum W₁ (v) is givenby

    W.sub.1 (v)=(1+b.sup.2 B(v).sup.2).sup.-1,

where b is a constant and B(v) is a positive, real valued function offrequency.
 32. The method of claim 30, wherein B(v) is proportional tov.sup.±n, where n is an integer.